A posteriori error estimates and adaptivity for locally conservative methods. Inexpensive implementation and evaluation, polytopal meshes, iterative linearization and algebraic solvers, and applications to complex porous media flows

Martin Vohral\'ik; Soleiman Yousef

REVIEW 3 major objections 4 minor 266 references

A posteriori error estimates and adaptivity for locally conservative methods. Inexpensive implementation and evaluation, polytopal meshes, iterative linearization and algebraic solvers, and applications to complex porous media flows

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For locally conservative methods on polytopal meshes, this paper proves guaranteed, fully computable error bounds that remain valid and cheap at each time step, linearization step, and algebraic solver step—culminating in multiphase…

desk verdict Careful, useful extension of the equilibrated-flux framework to polytopal meshes and the full solver chain, but the multiphase guarantee is certified residual control, not a proven distance to the exact solution. read the letter →

arxiv 2505.23245 v1 pith:U5CK3SOU submitted 2025-05-29 math.NA cs.NA

Martin Vohralík , Soleiman Yousef This is my paper

classification math.NAcs.NA MSC 65N1565N0865N3065M1576S05

keywords aposteriorierrorestimateslocallyconservativemethodsfinitevolumepolytopalmeshesequilibratedfluxreconstructioniterativelinearizationadaptivemeshrefinementmultiphaseflowinporousmedia

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to prove that a numerical simulation of complex porous-media flow can carry a certified, guaranteed upper bound on the distance between its approximate solution and the unknown exact solution at every stage of the resolution chain: each time step, each nonlinear-iteration step, and each linear-solver step. The authors build a posteriori error estimates for any lowest-order locally conservative method on general polytopal meshes, working through the Poisson equation, steady linear and nonlinear Darcy flow, and finally multiphase compositional Darcy flow. The estimates are fully computable from data already present in the code, evaluate by local matrix-vector products, and split the total error into spatial, temporal, linearization, algebraic, and remainder components. The stated payoff is adaptivity that stops nonlinear and linear solvers and refines or derefines the space and time meshes while keeping a guaranteed control of the global error. The flagship result, Theorem 7.6, bounds the intrinsic error at every stage of the multiphase simulation by a computable square root of a sum of squared component estimators.

What carries the argument

The load-bearing identity is the Prager–Synge equality, which expresses the squared flux error as the squared norm of a mismatch between a chosen $H(\mathrm{div},\Omega)$-conforming flux field and the gradient of a chosen $H^1_0(\Omega)$-conforming potential, minus the potential error term. Around this identity the paper builds two reconstructions: an $H(\mathrm{div},\Omega)$-conforming equilibrated flux reconstruction, obtained by lifting face normal fluxes into the lowest-order Raviart–Thomas space, and an $H^1_0(\Omega)$-conforming potential reconstruction, obtained by elementwise postprocessing and averaging. On polytopal meshes these are used only virtually: the estimator is evaluated directly through element stiffness, mass, and mixed-finite-element matrices acting on the face-flux and vertex-potential vectors, so no reconstruction or simplicial submesh is actually constructed. In the nonlinear and unsteady cases, the same matrices, multiplied by linearization and algebraic error face fluxes, yield the component estimators.

What would settle it

Choose a two- or three-phase compositional Darcy problem with a manufactured smooth solution on a moderately coarse polytopal mesh, fix one backward-Euler time step, and run several linearization and algebraic solver iterations; compute the exact intrinsic error $N^{n,k,i}$ from (7.17) and the estimator from (7.36) at every stage $(n,k,i)$. If any stage violates the inequality, the guaranteed-bound claim is refuted; on smooth problems the effectivity index should stay close to one from above.

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Extended reading notes

Core claim

The central discovery, stated in the language of the paper, is that the hypercircle machinery of Prager and Synge — in which the flux error is controlled by a mismatch between an $H(\mathrm{div},\Omega)$-conforming equilibrated flux reconstruction and an $H^1_0(\Omega)$-conforming potential reconstruction — extends through the entire resolution chain without ever constructing the reconstructions explicitly. For the multiphase compositional Darcy flow, Theorem 7.6, Eq. (7.36), asserts that at each time step $n$, linearization step $k$, and algebraic solver step $i$, \[ $N^{{n,k,i}}$\le \Big(\sum_{c\in C}\big(\eta_{\mathrm{sp},c}+\eta_{\mathrm{tm},c}+\eta_{\mathrm{lin},c}+\eta_{\mathrm{alg},c}+\eta_{\mathrm{rem},c}\big)^2\Big)^{1/2}, \] where the intrinsic error $N^{n,k,i}$ is defined in (7.17) as the dual norm of the residual plus the nonconformity of the phase fluxes. Each $\eta$ is assembled from elementwise estimators that are plain local matrix-vector products using the current algebraic unknowns and precomputed element matrices, and the different terms isolate spatial, temporal, upwinding, linearization, algebraic, and remainder errors. Thus the approximate solution at any intermediate stage of the solver carries an upper bound on its distance to the exact solution, conditional on the weak solution existing.

Load-bearing premise

Assumption 7.2 postulates existence, uniqueness, and sufficient regularity of a weak solution of the multiphase compositional model; if such a solution does not exist, the guaranteed upper bound of Theorem 7.6 has no exact solution to measure against, and the authors note that existence is currently proven only in simplified two-phase settings.

Editorial extensions

If this is right

An implementation can run adaptivity with guaranteed overall precision: stop the algebraic solver when $\eta_{\mathrm{alg}}\le\gamma_{\mathrm{alg}}\eta_{\mathrm{sp}}$, stop the linearization when $\eta_{\mathrm{lin}}\le\gamma_{\mathrm{lin}}\eta_{\mathrm{sp}}$, and refine or derefine in space and time to balance $\eta_{\mathrm{sp}}$ and $\eta_{\mathrm{tm}}$.
Even an inexact solve that is stopped early remains certified, because the same inequality is valid on every linearization and solver iteration, not only at convergence.
The evaluation cost of the estimators is the lowest possible order: only multiplications of precomputed element matrices by local face-flux and vertex-potential vectors, with no local problems solved.
Because all component estimators have the same units and the same form, the total error can be assigned to its four sources by a single computation, which is usually not possible with residual norms and iteration gaps.
The framework applies to any lowest-order locally conservative method on polytopal meshes, including finite volume, mixed finite element, mimetic finite difference, mixed virtual element, and hybrid high-order discretizations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: for reservoir simulation in practice, these component estimators give a direct way to replace heuristic solver tolerances with error-balance stopping criteria; the numerical experiments in the paper report sizable reduction of Newton iterations on this basis.
Editorial inference: if Assumption 7.2 holds, the same machinery could be extended toward goal-oriented certification, for example bounding the error in a cumulative production quantity, provided the quantity is Lipschitz with respect to the intrinsic error measure.
Editorial inference: higher-order locally conservative methods are a natural next test; the paper says the analysis carries over, so the matrix-vector form would likely survive with polynomial-degree-dependent element matrices.
Editorial inference: adaptive derefinement is delicate because estimators on coarsened cells must be built from a common simplicial refinement; the paper's Remark 7.8 indicates how, so a practical code must store the parent-child mesh relations between time steps.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a substantial, careful paper, and the multiphase section is where the central caveat lives. The genuinely new material is the extension to polytopal meshes via fictitious reconstructions, the matrix-vector evaluation of all estimators, and the full accounting of linearization and algebraic solver errors in the multiphase compositional model. Theorems 4.18, 5.18, 6.2, and 7.6 are proved with explicit constants and largely transparent assumptions, and the numerical experiments on polygonal meshes and the SPE10-type and black-oil cases support the practical claims.

The soft spot: Theorem 7.6 bounds the functional N^{n,k,i} defined in (7.17), and the paper calls that the intrinsic error measure. For the Poisson problem Theorem 4.9 proves equality between this functional and the true flux error; for the steady nonlinear case (6.33) gives equivalence. For the multiphase model no such equivalence is proved. So (7.36) is a certified residual/nonconformity bound, not a proven bound on the distance to the weak solution. The stress-test note is right about that. The authors are honest about Assumption 7.2 – existence and uniqueness of the multiphase weak solution is assumed, with known results only in much simpler settings – but that means the flagship guarantee is conditional on an open problem plus the missing equivalence.

Smaller caveats: the practical estimators in the experiments sometimes replace the guaranteed mixed-finite-element matrices with the scheme's own matrices, and the authors acknowledge this weakens the guarantee. No code or data is provided, and the real-life experiments are truncated, which hurts reproducibility. The heavy self-citation is justified because the framework genuinely builds on their earlier work.

Who this is for: numerical analysts working on a posteriori error control for finite-volume-type methods on polytopal meshes, and reservoir simulation practitioners who want adaptive stopping criteria for nonlinear and linear solvers. It deserves a serious referee; a good referee will ask for the Section 7 overclaim to be fixed – separate the certified residual bound from the certified error bound, or prove the missing equivalence – and for a reproducibility statement.

Referee Report

3 major / 4 minor

Summary. The paper develops a unified framework for guaranteed a posteriori error estimates for lowest-order locally conservative discretizations (finite volume type) on polytopal meshes. It treats successively the Poisson equation, steady linear Darcy flow, steady nonlinear Darcy flow with iterative linearization and algebraic solvers, and finally unsteady multiphase compositional Darcy flow with backward Euler time stepping. The estimators are designed so that their evaluation reduces to local matrix-vector products, with element matrices inherited from the scheme or built from geometry, and they distinguish spatial, temporal, linearization, and algebraic error components, enabling adaptive stopping and mesh/time-step adaptation. Main results are Theorems 4.18, 5.18, 6.2, and 7.6.

Significance. The paper is a substantial contribution to a posteriori error estimation for locally conservative methods. Its strengths are the explicit, computable constants in the linear and steady nonlinear estimates; the virtual reconstruction technique that avoids constructing simplicial submeshes in practice; and the unified treatment of solver errors, which gives practically useful adaptive stopping criteria. The numerical experiments, including three-dimensional reservoir-type cases, illustrate the methodology. However, the flagship result for multiphase flow (Theorem 7.6) currently bounds a residual/nonconformity functional rather than a proven distance to the weak solution, which weakens the 'guaranteed error bound' claim as stated.

major comments (3)

[Section 7.4, Eq. (7.17), and Theorem 7.6, Eq. (7.36)] The quantity N^{n,k,i} defined in (7.17) is a dual residual norm plus a nonconformity distance; it is not shown to be equivalent to any norm of X - X^{n,k,i}_{h\tau} (or of the corresponding flux error). For the Poisson case, Theorem 4.9 proves an exact characterization (4.9); for the steady nonlinear case, (6.33) proves two-sided control by the energy error. No such inequality is stated or proved for the multiphase system (7.1)-(7.12). Remark 7.4 asserts that N 'extends' Theorem 4.9 and Remark 6.6, but the coercivity/monotonicity/inf-sup argument needed for the degenerate coupled system is absent. Consequently, (7.36) bounds a computable residual/nonconformity functional, not the distance to the weak solution of Assumption 7.2. Since the abstract and Section 2.5.1 present the result as a guaranteed upper bound on the error between the unknown solution and the numerical approximation, this is a load-bearing gap: either supply the equivalence (under additional structural assumptions if necessary) or reformulate the claims as bounds on the intrinsic residual/nonconformity measure N^{n,k,i}.
[Section 7.9, Eq. (7.38b)] The nonconformity estimator for phase p is evaluated by formula (7.38b), which is derived in the proof of Theorem 5.18 from the identity (u_h, \nabla\zeta_h)_K = \langle u_h\cdot n, \zeta_h\rangle_{\partial K} - F_K |K|^{-1}(1,\zeta_h)_K, valid when \nabla\cdot u_h|_K = F_K/|K| (see (5.39)). For the Darcy phase flux reconstruction u^{n,k,i}_{p,h}, no such divergence property is stated. If the reconstruction is defined through Definition 5.5, the prescribed constant divergence must be identified for the face fluxes (7.32a); if instead a different lifting is used, the evaluation (7.38b) is unjustified. The proof of Theorem 7.6 should specify the reconstruction and verify the identity used.
[Assumption 7.2 and abstract] The main theorem for the real-life application is conditional on the existence and uniqueness of a weak solution to (7.1)-(7.12), which the authors explicitly note is open in the generality considered. This is an honest statement, but the abstract and introduction do not carry the same caveat: the 'guaranteed' bound in Theorem 7.6 is vacuous if the reference solution does not exist or is not unique. The authors should either state the theorem as conditional on Assumption 7.2 in the abstract or prove the equivalence/error bound under an additional assumption that is satisfied by the numerical benchmarks.

minor comments (4)

[Abstract] There is a typo in the abstract: 'mtehodology' should be 'methodology'.
[Theorem 7.6] The theorem statement says 'let the weak solution X satisfy (7.2)', but (7.2) only defines the vector X; the intended reference appears to be Assumption 7.2.
[Section 5.10] In the numerical experiment, the source term is not piecewise constant and the corresponding data-oscillation term is discarded; the text acknowledges this, but the figure captions should state that the displayed polygonal HFV bounds are not guaranteed in this test.
[Algorithm 7.1] The termination condition of Algorithm 7.1 contains typesetting artifacts ('T erminate' and malformed subscripts), which make it difficult to parse; please re-typeset the algorithm.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the estimators are derived and proved from residual/nonconformity functionals, with no fitted inputs and no self-citation chain forcing the result.

full rationale

The derivation chain is self-contained rather than circular. Sections 4–6 establish the framework: Theorem 4.9 proves an exact residual-plus-nonconformity characterization of the Poisson flux error, Theorem 5.18 proves a guaranteed bound using reconstructions evaluated by element matrices, and Theorem 6.2 proves a componentwise bound for nonlinear steady flow via monotonicity and the residual/nonconformity equivalence (6.33). Section 7 then defines the intrinsic error measure N^{n,k,i} in (7.17) precisely as a residual dual norm plus a nonconformity distance, and Theorem 7.6 proves that this measure is bounded by the computable estimators (7.38) through triangle inequalities, the Green theorem, and the reconstruction lemmas. No estimator is fitted to data; the element matrices are either inherited from the scheme or constructed from geometry, and the proof is included in the paper. The heavy self-citation is contextual rather than load-bearing: key ingredients such as Lemma 5.8 are proved here, and the cited prior work is not used as an unverified substitute for the present theorems. The main caveat is not circularity: Theorem 7.6 bounds the defined functional N, and Assumption 7.2 explicitly postulates existence and uniqueness of a weak solution, stating that proving it for the full multiphase model is too complicated. Whether N is equivalent to an actual distance to that weak solution is left open, as the paper itself concedes, but this is a correctness/completeness limitation, not a circular reduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 3 invented entities

The central claims rest on a standard functional-analytic background plus a set of clearly stated modeling and structural assumptions about the mesh, the data, and the numerical scheme. The most fragile item is Assumption 7.2 on the existence of a weak solution for the multiphase model. No parameter is fitted to data; all constants in the bounds are explicit (Poincare, Friedrichs, monotonicity constants) or user-chosen tolerances in the adaptive algorithm. The virtual submesh and the fictitious reconstructions are internal tools, not independently testable entities.

assumptions (6)

domain assumption Polytopal mesh TH admits a virtual simplicial submesh Th with uniform shape regularity (Section 3.6.1).
This structural assumption is used to define the Raviart-Thomas and piecewise-affine spaces used in the reconstructions and the element matrices bAMFE, bSFE, bMFE; the bounds' constants depend on the shape regularity.
domain assumption f and K are piecewise constant with respect to the polytopal mesh (Sections 5-7).
This ensures the reconstructed flux satisfies div uh = f exactly via the flux balance; without it, data oscillation terms appear, as acknowledged in Theorem 4.18.
domain assumption Assumption 5.2: the discrete saddle-point system has full-rank B, SPD element matrices bAK, and a lifting satisfying (5.11a)-(5.11c).
This covers the practical schemes (HFV, mimetic, mixed FV) and allows Corollaries 5.19-5.20 to use the scheme's own matrices directly. It is a property of the numerical method, not derived in the paper.
domain assumption Assumption 7.2: there exists a unique weak solution of the multiphase compositional model with regularity lc in Y, Pp in X, theta_c in L2 (Section 7.3).
The flagship guaranteed bound (Theorem 7.6) is meaningful only if the exact solution exists; the authors note existence is only known in simplified settings.
domain assumption Nonlinearity satisfies strong monotonicity and Lipschitz conditions (6.3)-(6.4) with known constants in Section 6.
These hypotheses on K are used to derive the constant-weighted error bound (6.24); for the multiphase model of Section 7 the proof instead relies on the intrinsic residual measure to avoid such constants.
standard math Standard functional analysis background: Sobolev spaces H^1_0 and H(div), Green's theorem, Riesz representation, Friedrichs/Poincare inequalities.
Invoked throughout Sections 3-7 in the proofs of the error characterizations; standard mathematical facts.

invented entities (3)

Virtual simplicial submesh Th of a polytopal mesh TH
purpose: Provides a common simplicial refinement on which H(div) and H^1-conforming reconstructions and element matrices are defined without being physically constructed.
The paper assumes such a submesh exists for every polytopal element (Section 3.6.1) but never constructs it in the algorithms; it is a theoretical device, not an observable entity.
Fictitious flux reconstruction u_h (Definitions 5.5, 6.6, Section 7.9)
purpose: A theoretical H(div)-conforming vector field used in the proofs and error definitions; not computed in the estimator evaluation.
It is defined from the scheme's face fluxes and used only to prove the bounds; its properties (divergence, face fluxes) are derived, not observed.
Fictitious potential reconstruction zeta_h (Definitions 5.14, 6.6, Section 7.9)
purpose: A theoretical H^1-conforming scalar field used in the Prager-Synge-type nonconformity estimates; the estimator only needs its nodal values from averaging formulas.
Similarly a proof device; the evaluable quantities are the nodal values Z_a and the stiffness/mass matrices.

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Cite this review

Pith. "Pith review of A posteriori error estimates and adaptivity for locally conservative methods. Inexpensive implementation and evaluation, polytopal meshes, iterative linearization and algebraic solvers, and applications to complex porous media flows." pith.science (2026). https://pith.science/paper/U5CK3SOU

@misc{pith2026250523245,
  author       = {Pith},
  title        = {Pith review of: A posteriori error estimates and adaptivity for locally conservative methods. Inexpensive implementation and evaluation, polytopal meshes, iterative linearization and algebraic solvers, and applications to complex porous media flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U5CK3SOU}},
  note         = {Machine review of arXiv:2505.23245}
}

read the original abstract

A posteriori estimates give bounds on the error between the unknown solution of a partial differential equation and its numerical approximation. We present here the methodology based on H1-conforming potential and H(div)-conforming equilibrated flux reconstructions, where the error bounds are guaranteed and fully computable. We consider any lowest-order locally conservative method of the finite volume type and treat general polytopal meshes. We start by a pure diffusion problem and first address the discretization error. We then progressively pass to more complicated model problems, up to complex multiphase multicomponent flow in porous media, and also take into account the errors arising in iterative linearization of nonlinear problems and in algebraic resolution of systems of linear algebraic equations. We focus on the ease of implementation and evaluation of the estimates. In particular, the evaluation of our estimates is explicit and inexpensive, since it merely consists in some local matrix-vector multiplications. Here, on each mesh element, the matrices are either directly inherited from the given numerical method, or easily constructed from the element geometry, while the vectors are the algebraic unknowns of the flux and potential approximations on the given element. Our mtehodology leads to an easy-to-implement and fast-to-run adaptive algorithm with guaranteed overall precision, adaptive stopping criteria for nonlinear and linear solvers, and adaptive space and time mesh refinements and derefinements. Progressively along the theoretical exposition, numerical experiments on academic benchmarks as well as on real-life problems in two and three space dimensions illustrate the performance of the derived methodology. The presentation is largely self-standing, developing all the details and recalling all necessary basic notions.

Figures

Figures reproduced from arXiv: 2505.23245 by the authors.

**Figure 2.** Example of a function belonging to the Sobolev space [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Simplicial mesh Th for d = 2 (left) and d = 3 (right) An illustration of a function v ∈ H1 (Th) is given in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (35 more)

**Figure 4.** Figure 4: Spaces Pk(K) and point values uniquely fixing a function vh ∈ Pk(K) (the so-called Lagrange nodes) for a mesh element K ∈ Th. Polynomial degrees k = 1 and k = 2, space dimensions d = 2 (left) and d = 3 (right) [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Spaces RT0(K) and normal fluxes uniquely fixing a function vh ∈ RT0(K) for a simplex K ∈ Th. Space dimensions d = 2 (left) and d = 3 (right) we write, in several equivalent ways, Pk(Th) ∩ H1 0 (Ω) = {vh ∈ H1 0 (Ω); vh|K ∈ Pk(K) ∀K ∈ Th} = {vh ∈ Pk(Th); (vh|K)|σ = (vh|L…

**Figure 6.** Figure 6: A basis function vσ of RT0(K) (supported on the simplex K) with nonzero normal flux only through the face σ ∈ FK (left). A basis function vσ of RT0(Th)∩H(div, Ω) (supported on all (two here) simplices sharing the face σ) with nonzero normal flux only through the face σ…

**Figure 7.** Figure 7: Polygonal element K ∈ TH (left). Virtual simplicial submesh TK of K (middle and right). One point inside K ∈ TH shared by all the simplices in TK and faces of Th do not subdividing the faces of TH (middle). General situation (right). where Qk(K) is the space of scalar-…

**Figure 8.** Figure 8: A polygonal mesh TH and the corresponding triangular submesh Th (P)K (UK)σ1 (UK)σ2 (UK)σ3 (UK)σ4 (UK)σ5 (UK)σ6 (UK)σ7 FK = {σi} 7 i=1 U ext K = {(UK)σi } 7 i=1 TK = {κi} 7 i=1 F ext K,h = {σi} 7 i=1 F int K,h = {σi} 14 i=8 FK,h = F ext K,h ∪ Fint K,h σ8 κ1 σ9 σ10 κ2 κ3…

**Figure 9.** Figure 9: Example of a polygonal element K with its faces FK, corresponding face fluxes U ext K , and pressure head (P)K (left); virtual simplicial submesh TK of K (right) faces of Th do not subdivide the faces of TH, then FH,h = FH and also F ext K,h = FK. For an interior face …

**Figure 10.** Figure 10: Admissible triangular mesh Th and notation for the cell-centered finite volume scheme UK,σ uh [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]

**Figure 11.** Figure 11: Finite volume face fluxes UK,σ (dashed black arrows) and flux reconstruction uh of Definition 4.14 (blue arrows) where UK,σ ∈ R for each face σ ∈ FK approximates the normal (out)flux ⟨u·nK, 1⟩σ from K over the face σ (recall the notation (3.12)) by UK,σ := − |σK,L| d…

**Figure 12.** Figure 12: Potential postprocessing ˜ph ∈ P2(Th) (left) and potential reconstruction by averaging ζh ∈ P2(Th) ∩ H1 (Ω) (right) Definition 4.17 (Potential reconstruction by averaging). Let p˜h ∈ P2(Th) be given by Definition 4.16. We call a potential reconstruction by averaging a…

**Figure 13.** Figure 13: [Section 4.11.1, mesh T3] The approximate solution given by the values pK, K ∈ Th, (left), the exact errors ∥u − uh∥K (middle), and the error estimators ηK from Theorem 4.18 (right). 104 105 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Number of unknowns Relative errors and es…

**Figure 14.** Figure 14: [Section 4.11.1, uniformly refined meshes T0–T3] Relative errors ∥u − uh∥/∥uh∥ and relative estimates η/∥uh∥ from Theorem 4.18 (left), effectivity indices Ieff from (4.29) (right) with the exact solution p(r, θ) = r 2 3 sin 2 3 θ + 3 2 π , where (r, θ) are the pol…

**Figure 15.** Figure 15: [Section 4.11.2, mesh T2] The approximate solution given by the values pK, K ∈ Th, (left), the exact errors ∥u − uh∥K (middle), and the error estimators ηK from Theorem 4.18 (right). 103 104 105 0.5 1 1.5 2 ·10−2 Number of unknowns Relative errors and estimators Error…

**Figure 16.** Figure 16: [Section 4.11.2, uniformly refined meshes T0–T2] Relative errors ∥u − uh∥/∥uh∥ and relative estimates η/∥uh∥ from Theorem 4.18 (left), effectivity indices Ieff from (4.29) (right) 5 Steady linear pure diffusion problems. Polytopal meshes and inexpensive implementation…

**Figure 17.** Figure 17: Face barycentres xK,σ and vertices aK,σ opposite to a face σ ∈ FK. Simplex K ∈ Th in space dimensions d = 2 (left) and d = 3 (right) Equivalently, writing the Euler–Lagrange conditions of (5.16) and imposing the divergence constraint with a Lagrange multiplier, uh|K ∈…

**Figure 18.** Figure 18: Example of element vectors ZK and Z ext K of nodal and facial potential point values where the latter condition is set in accordance with the boundary condition (5.1b). We will also need the face values {Zσ}σ∈FH,h Zσ := 1 d X a∈Vσ Za (5.26) for any simplicial face σ ∈…

**Figure 19.** Figure 19: The exact solution (5.43) cell K ∈ TH are constructed from the geometry of TK only, in the sense of Remark 5.16; 4) we use the estimator ηK (5.41) of Corollary 5.19 or 5.20. Note, however, that one may need a submesh of each polytopal cell K already to produce the sch…

**Figure 20.** Figure 20: Actual and estimated error distributions, entire domain ( [PITH_FULL_IMAGE:figures/full_fig_p041_20.png]

**Figure 21.** Figure 21: Relative error ∥u−uh∥ ∥uh∥ and relative estimators and η ∥uh∥ (left) and effectivity indices (right), uniform mesh refinement In [PITH_FULL_IMAGE:figures/full_fig_p041_21.png]

**Figure 22.** Figure 22: Relative error ∥u−uh∥ ∥uh∥ and relative estimators and η ∥uh∥ (left) and effectivity indices (right), adaptive mesh refinement leads to both smaller error and better effectivity indices in comparison with uniform mesh refinement. 5.11 Bibliographic resources Literatur…

**Figure 23.** Figure 23: Prototypical monotonous and Lipschitz-continuous function [PITH_FULL_IMAGE:figures/full_fig_p043_23.png]

**Figure 24.** Figure 24: [Section 6.8.1] The approximate solution given by the values p k,i K , K ∈ Th, (left) the energy errors c 1/2 K˜ ∥u−u k,i h ∥K (middle), and the error estimators η k,i sp,K +η k,i lin,K +η k,i alg,K +η k,i rem,K from Theorem 6.2 (right). 101 102 103 104 105 106 0.1 0.…

**Figure 25.** Figure 25: [Section 6.8.1], Relative errors c 1/2 K˜ ∥u − u k,i h ∥/∥u k,i h ∥ and relative estimates η k,i/∥u k,i h ∥ from Theorem 6.2 (left), effectivity indices Ieff from (6.34) (right) approximate solution given by the values p k,i K , K ∈ Th, the elementwise errors c 1/2 K˜…

**Figure 26.** Figure 26: [Section 6.8.1], Standard linearization vs. adaptive linearization [PITH_FULL_IMAGE:figures/full_fig_p052_26.png]

**Figure 27.** Figure 27: [Section 6.8.2] The approximate solution given by the values p k,i K , K ∈ Th, (left), the energy errors c 1/2 K˜ ∥u−u k,i h ∥K (middle), and the error estimators η k,i sp,K +η k,i lin,K +η k,i alg,K +η k,i rem,K from Theorem 6.2 (right). 6.8.2 Singular solution In th…

**Figure 28.** Figure 28: [Section 6.8.2], Relative errors c 1/2 K˜ ∥u − u k,i h ∥/∥u k,i h ∥ and relative estimates η k,i/∥u k,i h ∥ from Theorem 6.2 (left), effectivity indices Ieff from (6.34) (right) 0 5 10 15 10−8 10−6 10−4 10−2 100 adaptive stopping criterion standard stopping criterion …

**Figure 29.** Figure 29: [Section 6.8.2], Standard linearization vs. adaptive linearization given by the ratio of the estimate over the error (6.34). We see that the effectivity indices are not affected by the values of the ratio CK˜ /cK˜ , which numerically gives the robustness of the estima…

**Figure 30.** Figure 30: [Section 7.11.2] Porosity (left) and permeability (right), layer 85 of the 10th SPE case mesh in the sense of Section 3.6. Coarsening is then obtained by retrogressive cell agglomeration. During the coarsening and refinement phases, we use some upscaling and interpola…

**Figure 31.** Figure 31: [Section 7.11.2] Results on the fine mesh at 500 days the linear solver needed in Section 7.7, we use the Bi-Conjugated Gradient Stabilized (BiCGStab) [239] with an ILU{0} preconditioner [PITH_FULL_IMAGE:figures/full_fig_p065_31.png]

**Figure 32.** Figure 32: [Section 7.11.2] Results on the adaptive mesh at 400 days and 1100 days 0 0.5 1 1.5 ·108 0 1 2 3 4 ·104 Time (seconds) Cumulated oil rate ( m3 ) Fine mesh AMR Cartesian est. AMR polygonal est. Coarse mesh 0 0.5 1 1.5 ·108 0 1 2 3 ·105 Time (seconds) Water cut ( m3 ) F…

**Figure 33.** Figure 33: [Section 7.11.2] Fine mesh, coarse mesh, and adaptive meshes: cumulated oil rate (left) and water cut (right) dimensional test case leads to a gain factor in the overall CPU time at around 2, for both the polygonal estimate proposed in this paper and the Cartesian est…

**Figure 34.** Figure 34: [Section 7.11.3] Permeability 7.11.3 Three-phases, three-components Darcy flow In this section, we present a simulation of a black-oil model. Here, we have three phases constituted by water, oil, and gas, represented by lowercase letters w, o, g as indices, respective…

**Figure 35.** Figure 35: [Section 7.11.3] Results on the fine mesh at 1000 days, gas saturation (left) and normalized polygonal estimate (right) [PITH_FULL_IMAGE:figures/full_fig_p068_35.png]

**Figure 36.** Figure 36: [Section 7.11.3] Gas saturation: results on the adaptive mesh at 500 days (left) and 1500 days (right) [PITH_FULL_IMAGE:figures/full_fig_p068_36.png]

**Figure 37.** Figure 37: [Section 7.11.3] Standard resolution vs. adaptive resolution: total estimator and its algebraic component (left) and number of BiCGStab iterations per time step (right) 0 0.5 1 1.5 ·108 0 1 2 3 4 ·105 Time (seconds) Cumulated oil rate ( m3 ) Standard resolution Adapti…

**Figure 38.** Figure 38: [Section 7.11.3] Standard resolution vs. adaptive resolution: cumulated oil rate (left) and number of cells (right) We remark, in the left part of [PITH_FULL_IMAGE:figures/full_fig_p069_38.png]

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Works this paper leans on

266 extracted references · 89 canonical work pages

[1]

Discretization on unstructured grids for inhomogeneous, anisotropic media

Aavatsmark, I., Barkve, T., Bøe, Ø., and Mannseth, T. Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998), 1700–1716

1998
[2]

T., Klausen, R

Aavatsmark, I., Eigestad, G. T., Klausen, R. A., Wheeler, M. F., and Yotov, I. Convergence of a symmetric MPF A method on quadrilateral grids. Comput. Geosci. 11 (2007), 333–345. http: //dx.doi.org/10.1007/s10596-007-9056-8

work page doi:10.1007/s10596-007-9056-8 2007
[3]

A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations

Achdou, Y., Bernardi, C., and Coquel, F. A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations. Numer. Math. 96 (2003), 17–42. http://dx.doi.org/10. 1007/s00211-002-0436-7

2003
[4]

General purpose compositional model

´Acs, G., Doleschall, S., and Farkas, ´E. General purpose compositional model. Society of Petroleum Engineers Journal 25 (1985), 543–553. https://doi.org/10.2118/10515-PA

doi:10.2118/10515-pa 1985
[5]

Adams, R. A. Pure and Applied Mathematics, Vol. 65. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975

1975
[6]

A posteriori estimators for the finite vol- ume discretization of an elliptic problem

Afif, M., Bergam, A., Mghazli, Z., and Verf¨ urth, R. A posteriori estimators for the finite vol- ume discretization of an elliptic problem. Numer. Algorithms 34 (2003), 127–136. International Conference on Numerical Algorithms, Vol. II (Marrakesh, 2001)

2003
[7]

Connection between finite volume and mixed finite element methods for a diffusion problem with nonconstant coefficients

Agouzal, A., Baranger, J., Ma ˆ ıtre, J.-F., and Oudin, F. Connection between finite volume and mixed finite element methods for a diffusion problem with nonconstant coefficients. Application to a convection diffusion problem. East-West J. Numer. Math. 3 (1995), 237–254

1995
[8]

A posteriori error estimator for finite volume methods

Agouzal, A., and Oudin, F. A posteriori error estimator for finite volume methods. Appl. Math. Comput. 110 (2000), 239–250

2000

Show all 266 references

[9]

M., and Radu, F

Ahmed, E., Nordbotten, J. M., and Radu, F. A. Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems. J. Comput. Appl. Math. 364 (2020), 112312, 25. https://doi.org/10.1016/...

2020 doi
[10]

A., and Nordbotten, J

Ahmed, E., Radu, F. A., and Nordbotten, J. M. Adaptive poromechanics computations based on a posteriori error estimates for fully mixed formulations of Biot’s consolidation model. Comput. Methods Appl. Mech. Engrg. 347 (2019), 264–294. https://doi.org/10.1016/j.cma.2018.12. 016

2019 doi
[11]

Automatic simplification of Darcy’s equations with pressure dependent permeability

Ahusborde, E., Aza ¨ ıez, M., Ben Belgacem, F., and Bernardi, C. Automatic simplification of Darcy’s equations with pressure dependent permeability. ESAIM Math. Model. Numer. Anal. 47 (2013), 1797–1820. http://dx.doi.org/10.1051/m2an/2013089

2013
[12]

Robust a posteriori error estimation for nonconforming finite element approximation

Ainsworth, M. Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42 (2005), 2320–2341. https://doi.org/10.1137/S0036142903425112

2005 doi
[13]

A posteriori error estimation for lowest order Raviart–Thomas mixed finite elements

Ainsworth, M. A posteriori error estimation for lowest order Raviart–Thomas mixed finite elements. SIAM J. Sci. Comput. 30 (2007), 189–204. http://link.aip.org/link/?SCE/30/189/1

2007
[14]

Ainsworth, M., and Oden, J. T. A posteriori error estimation in finite element analysis . Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2000

2000
[15]

Analyse num´ erique et optimisation

Allaire, G. Analyse num´ erique et optimisation. Editions Ecole Polytechnique, Palaiseau, France, 2005

2005
[16]

W., and Luckhaus, S

Alt, H. W., and Luckhaus, S. Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311–341. http://dx.doi.org/10.1007/BF01176474

1983 doi
[17]

W., Luckhaus, S., and Visintin, A

Alt, H. W., Luckhaus, S., and Visintin, A. On nonstationary flow through porous media. Ann. Mat. Pura Appl. (4) 136 (1984), 303–316. http://dx.doi.org/10.1007/BF01773387

1984 doi
[18]

A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media

Amaziane, B., Bergam, A., El Ossmani, M., and Mghazli, Z. A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media. Internat. J. Numer. Methods Fluids 59 (2009), 259–284

2009
[19]

An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media

Amaziane, B., Jurak, M., and ˇZgalji´ c Keko, A. An existence result for a coupled system modeling a fully equivalent global pressure formulation for immiscible compressible two-phase flow in porous media. J. Differential Equations 250 (2011), 1685–1718. http://dx.doi.org/10.1...

2011 doi
[20]

Balanced a posteriori error estimates for finite-volume type discretizations of convection-dominated elliptic problems

Angermann, L. Balanced a posteriori error estimates for finite-volume type discretizations of convection-dominated elliptic problems. Computing 55 (1995), 305–323. https://doi.org/10. 1007/BF02238485

1995
[21]

An error estimator for a finite volume discretization of density driven flow in porous media

Angermann, L., Knabner, P., and Thiele, K. An error estimator for a finite volume discretization of density driven flow in porous media. Appl. Numer. Math. 26 (1998), 179–191. Proceedings of the International Centre for Mathematical Sciences Conference on Grid Adaptation in Co...

1998
[22]

F., Beir˜ ao da Veiga, L., Lovadina, C., and Verani, M

Antonietti, P. F., Beir˜ ao da Veiga, L., Lovadina, C., and Verani, M. Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems. SIAM J. Numer. Anal. 51 (2013), 654–675. http://dx.doi.org/10.1137/120873157

2013 doi
[23]

N., Kazhikhov, A

Antontsev, S. N., Kazhikhov, A. V., and Monakhov, V. N. Translated from the Russian. Boundary value problems in mechanics of nonhomogeneous fluids , vol. 22 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam, 1990

1990
[24]

A stopping criterion for the conjugate gradient algorithms in a finite element method framework

Arioli, M. A stopping criterion for the conjugate gradient algorithms in a finite element method framework. Numer. Math. 97 (2004), 1–24. http://dx.doi.org/10.1007/s00211-003-0500-y

2004 doi
[25]

H., and Loghin, D

Arioli, M., Georgoulis, E. H., and Loghin, D. Stopping criteria for adaptive finite element solvers. SIAM J. Sci. Comput. 35 (2013), A1537–A1559. http://dx.doi.org/10.1137/120867421

2013 doi
[26]

Interplay between discretization and algebraic computation in adaptive numerical solution of elliptic PDE problems

Arioli, M., Liesen, J., Mi ‘edlar, A., and Strakoˇ s, Z. Interplay between discretization and algebraic computation in adaptive numerical solution of elliptic PDE problems. GAMM-Mitt. 36 (2013), 102–129. http://dx.doi.org/10.1002/gamm.201310006

2013 doi
[27]

Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations

Axelsson, O., and Kaporin, I. Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations. Numer. Linear Algebra Appl. 8 (2001), 265–286. 71

2001
[28]

Petroleum Reservoir Simulation

Aziz, K., and Settari, A. Petroleum Reservoir Simulation. Applied Science Publishers, Ltd, London, 1979

1979
[29]

The finite element method and its reliability

Babuˇ ska, I., and Strouboulis, T. The finite element method and its reliability . Numerical Math- ematics and Scientific Computation. The Clarendon Press Oxford University Press, New York, 2001

2001
[30]

A posteriori stopping rule for regularized fixed point iterations

Bakushinsky, A., and Smirnova, A. A posteriori stopping rule for regularized fixed point iterations. Nonlinear Anal. 64 (2006), 1255–1261. http://dx.doi.org/10.1016/j.na.2005.06.031

2006 doi
[31]

Adaptive finite element methods for differential equations

Bangerth, W., and Rannacher, R. Adaptive finite element methods for differential equations . Lec- tures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, 2003.https://doi.org/10.1007/ 978-3-0348-7605-6

2003
[32]

Connection between finite volume and mixed finite element methods

Baranger, J., Ma ˆ ıtre, J.-F., and Oudin, F. Connection between finite volume and mixed finite element methods. RAIRO Mod´ el. Math. Anal. Num´ er.30 (1996), 445–465

1996
[33]

Exact a posteriori error control for variational problems via convex duality and explicit flux reconstruction

Bartels, S., and Kaltenbach, A. Exact a posteriori error control for variational problems via convex duality and explicit flux reconstruction. In Error control, adaptive discretizations, and applications. Part 1 , vol. 58 of Adv. Appl. Mech. Academic Press, San Diego, CA, 2024...

2024
[34]

Superconvergence and H(div) projection for discontinuous Galerkin methods

Bastian, P., and Rivi` ere, B. Superconvergence and H(div) projection for discontinuous Galerkin methods. Internat. J. Numer. Methods Fluids 42 (2003), 1043–1057

2003
[35]

Introduction to Modeling of Transport Phenomena in Porous Media , vol

Bear, J., and Bachmat, Y. Introduction to Modeling of Transport Phenomena in Porous Media , vol. 4 of Theory and Applications of Transport in Porous Media . Kluwer Academic Publishers, Dordrecht, Holland, 1990

1990
[36]

Non-isothermal compositional liquid gas Darcy flow: formulation, soil-atmosphere boundary condition and application to high- energy geothermal simulations

Beaude, L., Brenner, K., Lopez, S., Masson, R., and Smai, F. Non-isothermal compositional liquid gas Darcy flow: formulation, soil-atmosphere boundary condition and application to high- energy geothermal simulations. Comput. Geosci. 23 (2019), 443–470. https://doi.org/10.1007/...

2019
[37]

A note on the Poincar´ e inequality for convex domains

Bebendorf, M. A note on the Poincar´ e inequality for convex domains. Z. Anal. Anwendungen 22 (2003), 751–756. http://dx.doi.org/10.4171/ZAA/1170

2003 doi
[38]

Stopping criteria based on locally reconstructed fluxes

Becker, R., Capatina, D., and Luce, R. Stopping criteria based on locally reconstructed fluxes. In Numerical mathematics and advanced applications—ENUMATH 2013 , vol. 103 of Lect. Notes Comput. Sci. Eng. Springer, Cham, 2015, pp. 243–251

2013
[39]

Adaptive error control for multigrid finite element methods

Becker, R., Johnson, C., and Rannacher, R. Adaptive error control for multigrid finite element methods. Computing 55 (1995), 271–288. http://dx.doi.org/10.1007/BF02238483

1995 doi
[40]

D., and Russo, A

Beir˜ ao da Veiga, L., Brezzi, F., Marini, L. D., and Russo, A. Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM Math. Model. Numer. Anal. 50 (2016), 727–747. http://dx.doi.org/10.1051/m2an/2015067

2016
[41]

The mimetic finite difference method for elliptic problems, vol

Beir˜ ao da Veiga, L., Lipnikov, K., and Manzini, G. The mimetic finite difference method for elliptic problems, vol. 11 of MS&A. Modeling, Simulation and Applications . Springer, Cham, 2014. https://doi.org/10.1007/978-3-319-02663-3

2014 doi
[42]

A residual based error estimator for the mimetic finite difference method

Beir˜ ao da Veiga, L. A residual based error estimator for the mimetic finite difference method. Numer. Math. 108 (2008), 387–406. http://dx.doi.org/10.1007/s00211-007-0126-6

2008 doi
[43]

An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems

Beir˜ ao da Veiga, L., and Manzini, G. An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems. Internat. J. Numer. Methods Engrg. 76 (2008), 1696–1723. http://dx.doi.org/10.1002/nme.2377

2008 doi
[44]

A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints

Ben Gharbia, I., Dabaghi, J., Martin, V., and Vohral ´ ık, M. A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints. Comput. Geosci. 24 (2020), 1031–1055. https://doi.org/10.1007/s10596-019-09909-5

2020 doi
[45]

Semismooth and smoothing Newton meth- ods for nonlinear systems with complementarity constraints: Adaptivity and inexact resolution

Ben Gharbia, I., Ferzly, J., Vohral ´ ık, M., and Yousef, S. Semismooth and smoothing Newton meth- ods for nonlinear systems with complementarity constraints: Adaptivity and inexact resolution. J. Comput. Appl. Math. 420 (2023), 114765. https://doi.org/10.1016/j.cam.2022.114765. 72

2023
[46]

Analysis of non-isothermal multiphase flows in porous media

Beneˇ s, M. Analysis of non-isothermal multiphase flows in porous media. Math. Methods Appl. Sci. 45 (2022), 9653–9677. https://doi.org/10.1002/mma.8328

2022 doi
[47]

A posteriori analysis of the finite element discretization of some parabolic equations

Bergam, A., Bernardi, C., and Mghazli, Z. A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp. 74 (2005), 1117–1138

2005
[48]

Estimations a posteriori d’un sch´ ema de volumes finis pour un probl` eme non lin´ eaire.Numer

Bergam, A., Mghazli, Z., and Verf¨ urth, R. Estimations a posteriori d’un sch´ ema de volumes finis pour un probl` eme non lin´ eaire.Numer. Math. 95 (2003), 599–624

2003
[49]

A posteriori error analysis for two-overlapping domain decomposition techniques

Bernardi, C., Chac´ on Rebollo, T., Chac´ on Vera, E., and Franco Coronil, D. A posteriori error analysis for two-overlapping domain decomposition techniques. Appl. Numer. Math. 59 (2009), 1214–1236. http://dx.doi.org/10.1016/j.apnum.2008.06.004

2009 doi
[50]

A posteriori analysis of iterative algo- rithms for a nonlinear problem

Bernardi, C., Dakroub, J., Mansour, G., and Sayah, T. A posteriori analysis of iterative algo- rithms for a nonlinear problem. J. Sci. Comput. 65 (2015), 672–697. https://doi.org/10.1007/ s10915-014-9980-4

2015
[51]

A posteriori analysis of a space and time discretization of a nonlinear model for the flow in partially saturated porous media

Bernardi, C., El Alaoui, L., and Mghazli, Z. A posteriori analysis of a space and time discretization of a nonlinear model for the flow in partially saturated porous media. IMA J. Numer. Anal. 34 (2014), 1002–1036. http://dx.doi.org/10.1093/imanum/drt014

2014 doi
[52]

B., and Hyman, J

Bochev, P. B., and Hyman, J. M. Principles of mimetic discretizations of differential operators. In Compatible spatial discretizations, vol. 142 of IMA Vol. Math. Appl. Springer, New York, 2006, pp. 89–119. http://dx.doi.org/10.1007/0-387-38034-5_5

2006 doi
[53]

Mixed finite element methods and applications , vol

Boffi, D., Brezzi, F., and Fortin, M. Mixed finite element methods and applications , vol. 44 of Springer Series in Computational Mathematics . Springer, Heidelberg, 2013. https://doi.org/ 10.1007/978-3-642-36519-5

2013 doi
[54]

Boffi, D., and Di Pietro, D. A. Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes. ESAIM Math. Model. Numer. Anal. 52 (2018), 1–28. https: //doi.org/10.1051/m2an/2017036

2018
[55]

Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes

Bonelle, J., and Ern, A. Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM Math. Model. Numer. Anal. 48 (2014), 553–581. http://dx.doi.org/ 10.1051/m2an/2013104

2014
[56]

Hypercercle

Bossavit, A. “Hypercercle” et majorations d’erreur a posteriori par “corrections locales”. In ´Equations aux d´ eriv´ ees partielles et applications. Gauthier-Villars, ´Ed. Sci. M´ ed. Elsevier, Paris, 1998, pp. 221–237

1998
[57]

Equilibrated residual error estimator for edge elements

Braess, D., and Sch¨ oberl, J. Equilibrated residual error estimator for edge elements. Math. Comp. 77 (2008), 651–672. http://dx.doi.org/10.1090/S0025-5718-07-02080-7

2008 doi
[58]

A cell-centered diffusion scheme on two-dimensional unstructured meshes

Breil, J., and Maire, P.-H. A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comput. Phys. 224 (2007), 785–823. http://dx.doi.org/10.1016/j.jcp.2006.10.025

2007 doi
[59]

Sequential implicit vertex approximate gradient dis- cretization of incompressible two-phase Darcy flows with discontinuous capillary pressure

Brenner, K., Chorfi, N., and Masson, R. Sequential implicit vertex approximate gradient dis- cretization of incompressible two-phase Darcy flows with discontinuous capillary pressure. Comput. Geosci. 26 (2022), 147–169. https://doi.org/10.1007/s10596-021-10113-7

2022 doi
[60]

H., and Droniou, J

Brenner, K., Masson, R., Quenjel, E. H., and Droniou, J. Total velocity-based finite volume dis- cretization of two-phase Darcy flow in highly heterogeneous media with discontinuous capillary pres- sure. IMA J. Numer. Anal. 42 (2022), 1231–1272. https://doi.org/10.1093/imanum/drab018

2022 doi
[61]

C., and Scott, L

Brenner, S. C., and Scott, L. R. The mathematical theory of finite element methods , third ed., vol. 15 of Texts in Applied Mathematics . Springer, New York, 2008. http://dx.doi.org/10. 1007/978-0-387-75934-0

2008
[62]

S., and Marini, L

Brezzi, F., Falk, R. S., and Marini, L. D. Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48 (2014), 1227–1240. https://doi.org/10.1051/m2an/2013138

2014
[63]

Mixed and hybrid finite element methods , vol

Brezzi, F., and Fortin, M. Mixed and hybrid finite element methods , vol. 15 of Springer Series in Computational Mathematics . Springer-Verlag, New York, 1991. http://dx.doi.org/10.1007/ 978-1-4612-3172-1 . 73

1991
[64]

Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes

Brezzi, F., Lipnikov, K., and Shashkov, M. Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005), 1872–1896

2005
[65]

A family of mimetic finite difference methods on polygonal and polyhedral meshes

Brezzi, F., Lipnikov, K., and Simoncini, V. A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15 (2005), 1533–1551. http: //dx.doi.org/10.1142/S0218202505000832

2005 doi
[66]

On full linear conver- gence and optimal complexity of adaptive FEM with inexact solver

Bringmann, P., Feischl, M., Mira¸ ci, A., Praetorius, D., and Streitberger, J. On full linear conver- gence and optimal complexity of adaptive FEM with inexact solver. Comput. Math. Appl. 180 (2025), 102–129. https://doi.org/10.1016/j.camwa.2024.12.013

2025 doi
[67]

ArbiLoMod, a simulation technique designed for arbitrary local modifications

Buhr, A., Engwer, C., Ohlberger, M., and Rave, S. ArbiLoMod, a simulation technique designed for arbitrary local modifications. SIAM J. Sci. Comput. 39 (2017), A1435–A1465. https://doi. org/10.1137/15M1054213

2017 doi
[68]

Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations

Burman, E., and Ern, A. Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations. Math. Comp. 76 (2007), 1119–1140

2007
[69]

Two-phase flows involving capillary barriers in hetero- geneous porous media

Canc` es, C., Gallou¨ et, T., and Porretta, A. Two-phase flows involving capillary barriers in hetero- geneous porous media. Interfaces Free Bound. 11 (2009), 239–258. http://dx.doi.org/10.4171/ IFB/210

2009
[70]

S., and Vohral ´ ık, M

Canc` es, C., Pop, I. S., and Vohral ´ ık, M. An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. Math. Comp. 83 (2014), 153–

2014
[71]

Cao, X., and Pop, I. S. Two-phase porous media flows with dynamic capillary effects and hysteresis: uniqueness of weak solutions. Comput. Math. Appl. 69 (2015), 688–695. https://doi.org/10. 1016/j.camwa.2015.02.009

2015
[72]

Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository

Caro, F., Saad, B., and Saad, M. Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository. Discrete Contin. Dyn. Syst. Ser. S 7 (2014), 191–205. https://doi.org/10.3934/dcdss.2014.7.191

2014 doi
[73]

Explicit and averaging a posteriori error estimates for adaptive finite volume methods

Carstensen, C., Lazarov, R., and Tomov, S. Explicit and averaging a posteriori error estimates for adaptive finite volume methods. SIAM J. Numer. Anal. 42 (2005), 2496–2521

2005
[74]

A posteriori estimation of the linearization error for strongly monotone nonlinear operators

Chaillou, A., and Suri, M. A posteriori estimation of the linearization error for strongly monotone nonlinear operators. J. Comput. Appl. Math. 205 (2007), 72–87. http://dx.doi.org/10.1016/ j.cam.2006.04.041

2007
[75]

L., and Suri, M

Chaillou, A. L., and Suri, M. Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Engrg. 196 (2006), 210–224. http: //dx.doi.org/10.1016/j.cma.2006.03.008

2006 doi
[76]

An introductory review on a posteriori error estimation in finite element computations

Chamoin, L., and Legoll, F. An introductory review on a posteriori error estimation in finite element computations. SIAM Rev. 65 (2023), 963–1028. https://doi.org/10.1137/21M1464841

2023 doi
[77]

H., Estep, D., and Tavener, S

Chaudhry, J. H., Estep, D., and Tavener, S. J. A posteriori error analysis for Schwarz overlap- ping domain decomposition methods. BIT 61 (2021), 1153–1191. https://doi.org/10.1007/ s10543-021-00864-1

2021
[78]

Studies in Mathematics and Its Applications, Vol

Chavent, G., and Jaffr´ e, J. Studies in Mathematics and Its Applications, Vol. 17. Mathematical models and finite elements for reservoir simulation . North-Holland, Amsterdam, 1986

1986
[79]

On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles

Chavent, G., Youn` es, A., and Ackerer, P. On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles. Comput. Methods Appl. Mech. Engrg. 192 (2003), 655–682

2003
[80]

A posteriori error estimates of mixed methods for miscible displacement problems

Chen, Y., and Liu, W. A posteriori error estimates of mixed methods for miscible displacement problems. Internat. J. Numer. Methods Engrg. 73 (2008), 331–343. http://dx.doi.org/10.1002/ nme.2075. 74

2008
[81]

Degenerate two-phase incompressible flow

Chen, Z. Degenerate two-phase incompressible flow. I. Existence, uniqueness and regularity of a weak solution. J. Differential Equations 171 (2001), 203–232. http://dx.doi.org/10.1006/jdeq. 2000.3848

2001
[82]

Degenerate two-phase incompressible flow

Chen, Z. Degenerate two-phase incompressible flow. II. Regularity, stability and stabilization. J. Differential Equations 186 (2002), 345–376. http://dx.doi.org/10.1016/S0022-0396(02) 00027-X

2002 doi
[83]

Mathematical techniques in oil recovery

Chen, Z. Mathematical techniques in oil recovery. Reservoir simulation, vol. 77 of CBMS-NSF Re- gional Conference Series in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. https://doi.org/10.1137/1.9780898717075

2007 doi
[84]

Chen, Z., and Ewing, R. E. Degenerate two-phase incompressible flow. III. Sharp error estimates. Numer. Math. 90 (2001), 215–240. http://dx.doi.org/10.1007/s002110100291

2001 doi
[85]

Chen, Z., and Ewing, R. E. Degenerate two-phase incompressible flow. IV. Local refinement and domain decomposition. J. Sci. Comput. 18 (2003), 329–360. http://dx.doi.org/10.1023/A: 1022673427893

2003 doi
[86]

Computational methods for multiphase flows in porous media

Chen, Z., Huan, G., and Ma, Y. Computational methods for multiphase flows in porous media . Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. http://dx.doi.org/10.1137/1.9780898718942

2006 doi
[87]

Christiansen, S. H. A construction of spaces of compatible differential forms on cellular com- plexes. Math. Models Methods Appl. Sci. 18 (2008), 739–757. http://dx.doi.org/10.1142/ S021820250800284X

2008
[88]

Christiansen, S. H. ´El´ ements finis mixtes minimaux sur les poly` edres.C. R. Math. Acad. Sci. Paris 348 (2010), 217–221. http://dx.doi.org/10.1016/j.crma.2010.01.017

2010 doi
[89]

A., and Blunt, M

Christie, M. A., and Blunt, M. J. Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reservoir Eval. Eng. 4 (2001), 308–317. https://doi.org/10.2118/ 72469-PA

2001
[90]

An h-adaptive operator splitting method for two- phase flow in 3D heterogeneous porous media

Chueh, C.-C., Djilali, N., and Bangerth, W. An h-adaptive operator splitting method for two- phase flow in 3D heterogeneous porous media. SIAM J. Sci. Comput. 35 (2013), B149–B175. https://doi.org/10.1137/120866208

2013 doi
[91]

C., Secanell, M., Bangerth, W., and Djilali, N

Chueh, C. C., Secanell, M., Bangerth, W., and Djilali, N. Multi-level adaptive simulation of transient two-phase flow in heterogeneous porous media. Comput. & Fluids 39 (2010), 1585–1596. https://doi.org/10.1016/j.compfluid.2010.05.011

2010 doi
[92]

Ciarlet, P. G. The Finite Element Method for Elliptic Problems , vol. 4 of Studies in Mathematics and its Applications . North-Holland, Amsterdam, 1978

1978
[93]

Coats, K. H. An equation of state compositional model. Society of Petroleum Engineers Journal 20 (1980), 363–376. https://doi.org/10.2118/8284-PA

1980 doi
[94]

Coats, K. H. Implicit compositional simulation of single-porosity and dual-porosity reservoirs. In SPE Symposium on Reservoir Simulation (1989). Paper SPE-18427-MS, https://doi.org/10. 2118/18427-MS

1989
[95]

A., and Ern, A

Cockburn, B., Di Pietro, D. A., and Ern, A. Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM Math. Model. Numer. Anal. 50 (2016), 635–650. http: //dx.doi.org/10.1051/m2an/2015051

2016
[96]

Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method

Cottereau, R., D ´ ıez, P., and Huerta, A. Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method. Comput. Mech. 44 (2009), 533–547. https://doi. org/10.1007/s00466-009-0388-1

2009 doi
[97]

Explicit error bounds in a conforming finite element method

Destuynder, P., and M´ etivet, B. Explicit error bounds in a conforming finite element method. Math. Comp. 68 (1999), 1379–1396. http://dx.doi.org/10.1090/S0025-5718-99-01093-5 . 75

1999 doi
[98]

H., Franc, J., Mø yner, O., and Tchelepi, H

Deucher, R. H., Franc, J., Mø yner, O., and Tchelepi, H. A. Compositional reservoir simulation with a high-resolution compact stencil adaptive implicit method. J. Comput. Phys. 521 (2025), Paper No. 113558, 17. https://doi.org/10.1016/j.jcp.2024.113558

2025
[99]

Adaptive numerical solution of PDEs

Deuflhard, P., and Weiser, M. Adaptive numerical solution of PDEs . de Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012. http://dx.doi.org/10.1515/9783110283112

2012 doi
[100]

A., and Ern, A

Di Pietro, D. A., and Ern, A. Mathematical aspects of discontinuous Galerkin methods , vol. 69 of Math´ ematiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-22980-0

2012 doi
[101]

A., and Ern, A

Di Pietro, D. A., and Ern, A. A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Engrg. 283 (2015), 1–21. http://dx.doi.org/10. 1016/j.cma.2014.09.009

2015
[102]

A., Ern, A., and Lemaire, S

Di Pietro, D. A., Ern, A., and Lemaire, S. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14 (2014), 461–472. https://doi.org/10.1515/cmam-2014-0018

2014 doi
[103]

A., Flauraud, E., Vohral ´ ık, M., and Yousef, S

Di Pietro, D. A., Flauraud, E., Vohral ´ ık, M., and Yousef, S. A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media. J. Comput. Phys. 276 (2014), 163–187. http://dx.doi.org/10.1016/j.jcp.2014.06.061

2014 doi
[104]

A., Vohral ´ ık, M., and Yousef, S

Di Pietro, D. A., Vohral ´ ık, M., and Yousef, S. An a posteriori-based, fully adaptive algorithm with adaptive stopping criteria and mesh refinement for thermal multiphase compositional flows in porous media. Comput. Math. Appl. 68 (2014), 2331–2347. https://doi.org/10.1016/j...

2014 doi
[105]

Use of corner point geometry in reservoir simulation

Ding, Y., and Lemonnier, P. Use of corner point geometry in reservoir simulation. In International Meeting on Petroleum Engineering (1995). Paper SPE-29933-MS, https://doi.org/10.2118/ 29933-MS

1995
[106]

Guaranteeda posteriori error bounds for low-rank tensor approxi- mate solutions

Dolgov, S., and Vejchodsk´ y, T. Guaranteeda posteriori error bounds for low-rank tensor approxi- mate solutions. IMA J. Numer. Anal. 41 (2021), 1240–1266. https://doi.org/10.1093/imanum/ draa010

2021 doi
[107]

Finite volume schemes for diffusion equations: introduction to and review of modern methods

Droniou, J. Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24 (2014), 1575–1619. http://dx.doi.org/10.1142/ S0218202514400041

2014
[108]

A mixed finite volume scheme for anisotropic diffusion problems on any grid

Droniou, J., and Eymard, R. A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105 (2006), 35–71

2006
[109]

The gradient discretisation method, vol

Droniou, J., Eymard, R., Gallou¨ et, T., Guichard, C., and Herbin, R. The gradient discretisation method, vol. 82 of Math´ ematiques & Applications (Berlin) [Mathematics & Applications]. Springer, Cham, 2018. https://doi.org/10.1007/978-3-319-79042-8

2018 doi
[110]

A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods

Droniou, J., Eymard, R., Gallou¨ et, T., and Herbin, R. A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20 (2010), 265–295. http://dx.doi.org/10.1142/S0218202510004222

2010 doi
[111]

Gradient schemes: generic tools for the numerical analysis of diffusion equations

Droniou, J., Eymard, R., and Herbin, R. Gradient schemes: generic tools for the numerical analysis of diffusion equations. ESAIM Math. Model. Numer. Anal. 50 (2016), 749–781. http://dx.doi. org/10.1051/m2an/2015079

2016
[112]

An error estimator for nonconforming approximations of a nonlinear problem

Dur´ an, R., and Padra, C. An error estimator for nonconforming approximations of a nonlinear problem. In Finite element methods (Jyv¨ askyl¨ a, 1993), vol. 164 of Lecture Notes in Pure and Appl. Math. Dekker, New York, 1994, pp. 201–205

1993
[113]

Edwards, M. G. Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6 (2002), 433–452. Locally conservative numerical methods for flow in porous media, http://dx.doi.org/10.1023/A:1021243231313. 76

2002 doi
[114]

Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems.Comput

El Alaoui, L., Ern, A., and Vohral ´ ık, M. Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems.Comput. Methods Appl. Mech. Engrg. 200 (2011), 2782–2795. http://dx.doi.org/10.1016/j.cma.2010.03.024

2011 doi
[115]

Mesh adaptation for two-phase flow

El Hassouni, S., and Mghazli, Z. Mesh adaptation for two-phase flow. Int. J. Math. Stat. 13 (2013), 56–68. https://doi.org/10.1002/pamm.201310023

2013 doi
[116]

A posteriori error estimate and adaptive mesh refinement for the cell-centered finite volume method for elliptic boundary value problems

Erath, C., and Praetorius, D. A posteriori error estimate and adaptive mesh refinement for the cell-centered finite volume method for elliptic boundary value problems. SIAM J. Numer. Anal. 47 (2008/09), 109–135. http://dx.doi.org/10.1137/070702126

2008 doi
[117]

Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems

Eriksson, K., and Johnson, C. Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comp. 60 (1993), 167–188, S1–S2

1993
[118]

Finite Elements I

Ern, A., and Guermond, J.-L. Finite Elements I. Approximation and Interpolation , vol. 72 of Texts in Applied Mathematics. Springer International Publishing, Springer Nature Switzerland AG, 2021. https://doi-org/10.1007/978-3-030-56341-7

2021 doi
[119]

An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems

Ern, A., Nicaise, S., and Vohral ´ ık, M. An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C. R. Math. Acad. Sci. Paris 345 (2007), 709–712. http://dx.doi.org/10.1016/j.crma.2007.10.036

2007 doi
[120]

Guaranteed, locally space-time efficient, and polynomial- degree robust a posteriori error estimates for high-order discretizations of parabolic problems.SIAM J

Ern, A., Smears, I., and Vohral ´ ık, M. Guaranteed, locally space-time efficient, and polynomial- degree robust a posteriori error estimates for high-order discretizations of parabolic problems.SIAM J. Numer. Anal. 55 (2017), 2811–2834. https://doi.org/10.1137/16M1097626

2017 doi
[121]

Equilibrated flux a posteriori error estimates inL2(H 1)-norms for high-order discretizations of parabolic problems

Ern, A., Smears, I., and Vohral ´ ık, M. Equilibrated flux a posteriori error estimates inL2(H 1)-norms for high-order discretizations of parabolic problems. IMA J. Numer. Anal. 39 (2019), 1158–1179. https://doi.org/10.1093/imanum/dry035

2019 doi
[122]

F., and Vohral ´ ık, M

Ern, A., Stephansen, A. F., and Vohral ´ ık, M. Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems. J. Comput. Appl. Math. 234 (2010), 114–130. http://dx.doi.org/10.1016/j.cam.2009.12.009

2010 doi
[123]

A posteriori error estimation based on potential and flux reconstruction for the heat equation

Ern, A., and Vohral ´ ık, M. A posteriori error estimation based on potential and flux reconstruction for the heat equation. SIAM J. Numer. Anal. 48 (2010), 198–223. http://dx.doi.org/10.1137/ 090759008

2010
[124]

Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs

Ern, A., and Vohral ´ ık, M. Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. 35 (2013), A1761–A1791. http://dx.doi. org/10.1137/120896918

2013 doi
[125]

Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations

Ern, A., and Vohral ´ ık, M. Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53 (2015), 1058–1081. http://dx.doi.org/10.1137/130950100

2015 doi
[126]

Finite volume methods

Eymard, R., Gallou¨ et, T., and Herbin, R. Finite volume methods. In Handbook of Numerical Analysis, Vol. VII . North-Holland, Amsterdam, 2000, pp. 713–1020

2000
[127]

Finite volume approximation of elliptic problems and convergence of an approximate gradient

Eymard, R., Gallou¨ et, T., and Herbin, R. Finite volume approximation of elliptic problems and convergence of an approximate gradient. Appl. Numer. Math. 37 (2001), 31–53. http://dx.doi. org/10.1016/S0168-9274(00)00024-6

2001 doi
[128]

Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid inter- faces

Eymard, R., Gallou¨ et, T., and Herbin, R. Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid inter- faces. IMA J. Numer. Anal. 30 (2010), 1009–1043. http://dx.doi.org/10.1093/imanum/drn084

2010 doi
[129]

Vertex-centred discretization of multiphase compositional Darcy flows on general meshes

Eymard, R., Guichard, C., Herbin, R., and Masson, R. Vertex-centred discretization of multiphase compositional Darcy flows on general meshes. Comput. Geosci. 16 (2012), 987–1005

2012
[130]

A., Helmig, R., Becker, B., Flemisch, B., and Geiger, S

Faigle, B., Elfeel, M. A., Helmig, R., Becker, B., Flemisch, B., and Geiger, S. Multi-physics modeling of non-isothermal compositional flow on adaptive grids. Comput. Methods Appl. Mech. Engrg. 292 (2015), 16–34. https://doi.org/10.1016/j.cma.2014.11.030. 77

2015 doi
[131]

Efficient multiphysics modelling with adaptive grid refinement using a MPF A method

Faigle, B., Helmig, R., Aavatsmark, I., and Flemisch, B. Efficient multiphysics modelling with adaptive grid refinement using a MPF A method. Comput. Geosci. 18 (2014), 625–636. https: //doi.org/10.1007/s10596-014-9407-1

2014 doi
[132]

A control volume method to solve an elliptic equation on a two-dimensional irregular mesh

Faille, I. A control volume method to solve an elliptic equation on a two-dimensional irregular mesh. Comput. Methods Appl. Mech. Engrg. 100 (1992), 275–290

1992
[133]

Mathematical analysis of variable density flows in porous media

Feireisl, E., Hilhorst, D., Petzeltov´ a, H., and Tak´ aˇ c, P. Mathematical analysis of variable density flows in porous media. J. Evol. Equ. 16 (2016), 1–19. https://doi.org/10.1007/ s00028-015-0290-6

2016
[134]

Adaptive regularization, discretization, and lineariza- tion for nonsmooth problems based on primal-dual gap estimators

F´ evotte, F., Rappaport, A., and Vohral ´ ık, M. Adaptive regularization, discretization, and lineariza- tion for nonsmooth problems based on primal-dual gap estimators. Comput. Methods Appl. Mech. Engrg. 418 (2024), 116558. https://doi.org/10.1016/j.cma.2023.116558

2024
[135]

Ganis, B., Pencheva, G., and Wheeler, M. F. Adaptive mesh refinement with an enhanced velocity mixed finite element method on semi-structured grids using a fully coupled solver. Comput. Geosci. 23 (2019), 149–168. https://doi.org/10.1007/s10596-018-9789-6

2019 doi
[136]

Rate optimality of adaptive finite element methods with respect to overall computational costs

Gantner, G., Haberl, A., Praetorius, D., and Schimanko, S. Rate optimality of adaptive finite element methods with respect to overall computational costs. Math. Comp. 90 (2021), 2011–2040. https://doi.org/10.1090/mcom/3654

2021 doi
[137]

H., Lakkis, O., and Virtanen, J

Georgoulis, E. H., Lakkis, O., and Virtanen, J. M. A posteriori error control for discontinuous Galerkin methods for parabolic problems. SIAM J. Numer. Anal. 49 (2011), 427–458. http: //dx.doi.org/10.1137/080722461

2011 doi
[138]

H., Lakkis, O., and Wihler, T

Georgoulis, E. H., Lakkis, O., and Wihler, T. P. A posteriori error bounds for fully-discrete hp- discontinuous Galerkin timestepping methods for parabolic problems. Numer. Math. 148 (2021), 363–386. https://doi.org/10.1007/s00211-021-01187-7

2021 doi
[139]

H., and Makridakis, C

Georgoulis, E. H., and Makridakis, C. G. Lower bounds, elliptic reconstruction and a posteriori error control of parabolic problems. IMA J. Numer. Anal. 43 (2023), 3212–3242. https://doi. org/10.1093/imanum/drac080

2023 doi
[140]

H., and Strakoˇ s, Z

Golub, G. H., and Strakoˇ s, Z. Estimates in quadratic formulas. Numer. Algorithms 8 (1994), 241–268. http://dx.doi.org/10.1007/BF02142693

1994 doi
[141]

The Arcane development framework

Grospellier, G., and Lelandais, B. The Arcane development framework. In Proceedings of the 8th workshop on Parallel/High-Performance Object-Oriented Scientific Computing (New York, NY, USA, 2009), POOSC ’09, ACM, pp. 4:1–4:11. http://doi.acm.org/10.1145/1595655.1595659

2009
[142]

Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver

Haberl, A., Praetorius, D., Schimanko, S., and Vohral ´ ık, M. Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver. Numer. Math. 147 (2021), 679–725. https://doi.org/10.1007/s00211-021-01176-w

2021 doi
[143]

A posteriori error analysis for linearization of nonlinear elliptic problems and their dis- cretizations

Han, W. A posteriori error analysis for linearization of nonlinear elliptic problems and their dis- cretizations. Math. Methods Appl. Sci. 17 (1994), 487–508. http://dx.doi.org/10.1002/mma. 1670170702

1994 doi
[144]

With applications in modeling and numerical approximations

Han, W. With applications in modeling and numerical approximations. A posteriori error analysis via duality theory , vol. 8 of Advances in Mechanics and Mathematics . Springer-Verlag, New York, 2005

2005
[145]

Robust augmented energy a posteriori estimates for Lipschitz and strongly monotone elliptic problems

Harnist, A., Mitra, K., Rappaport, A., and Vohral ´ ık, M. Robust augmented energy a posteriori estimates for Lipschitz and strongly monotone elliptic problems. HAL Preprint 04033438, https: //hal.inria.fr/hal-04033438, 2024

2024
[146]

Heid, P., Praetorius, D., and Wihler, T. P. Energy contraction and optimal convergence of adaptive iterative linearized finite element methods. Comput. Methods Appl. Math. 21 (2021), 407–422. https://doi.org/10.1515/cmam-2021-0025

2021 doi
[147]

Heid, P., and Wihler, T. P. Adaptive iterative linearization Galerkin methods for nonlinear prob- lems. Math. Comp. 89 (2020), 2707–2734. https://doi.org/10.1090/mcom/3545. 78

2020 doi
[148]

Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media

Henning, P., Ohlberger, M., and Schweizer, B. Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media. Comput. Geosci. 19 (2015), 99–114. https://doi. org/10.1007/s10596-014-9455-6

2015 doi
[149]

An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh

Herbin, R. An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods Partial Differential Equations 11 (1995), 165–173. https: //doi.org/10.1002/num.1690110205

1995 doi
[150]

Benchmark on discretization schemes for anisotropic diffusion problems on general grids

Herbin, R., and Hubert, F. Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In Finite volumes for complex applications V . ISTE, London, 2008, pp. 659–692

2008
[151]

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

Hilhorst, D., and Vohral ´ ık, M. A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids. Comput. Methods Appl. Mech. Engrg. 200 (2011), 597–613. http://dx.doi.org/10.1016/j.cma.2010.08.017

2011 doi
[152]

Translated from the Slovak by J

Hlav´ aˇ cek, I., Haslinger, J., Neˇ cas, J., and Lov ´ ıˇ sek, J. Translated from the Slovak by J. Jarn ´ ık.So- lution of variational inequalities in mechanics , vol. 66 of Applied Mathematical Sciences. Springer- Verlag, New York, 1988

1988
[153]

H., and Tchelep, H

Jenny, P., Lee, S. H., and Tchelep, H. A. Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Model. Simul. 3 (2004/05), 50–64. http://dx.doi. org/10.1137/030600795

2004 doi
[154]

A posteriori error estimates including algebraic error and stopping criteria for iterative solvers

Jir´ anek, P., Strakoˇ s, Z., and Vohral ´ ık, M. A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32 (2010), 1567–1590. http: //dx.doi.org/10.1137/08073706X

2010 doi
[155]

S., and S¨ uli, E

Jovanovi´ c, B. S., and S¨ uli, E. For linear partial differential equations with generalized solutions. Analysis of finite difference schemes , vol. 46 of Springer Series in Computational Mathematics . Springer, London, 2014. https://doi.org/10.1007/978-1-4471-5460-0

2014 doi
[156]

A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems

Kaltenbacher, B. A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems. Numer. Math. 79 (1998), 501–528. http: //dx.doi.org/10.1007/s002110050349

1998 doi
[157]

A., and Pascal, F

Karakashian, O. A., and Pascal, F. A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003), 2374–2399. http://dx.doi.org/10.1137/S0036142902405217

2003 doi
[158]

Keilegavlen, E., and Nordbotten, J. M. Conservative inexact solvers for porous media flow. In Domain decomposition methods in science and engineering XXI , vol. 98 of Lect. Notes Comput. Sci. Eng. Springer, Cham, 2014, pp. 333–340

2014
[159]

Keilegavlen, E., and Nordbotten, J. M. Inexact linear solvers for control volume discretiza- tions in porous media. Comput. Geosci. 19 (2015), 159–176. https://doi.org/10.1007/ s10596-014-9453-8

2015
[160]

Kelley, C. T. With separately available software. Iterative methods for linear and nonlinear equa- tions, vol. 16 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. https://doi.org/10.1137/1.9781611970944

1995 doi
[161]

Degenerate two-phase compressible immiscible flow in porous media: the case where the density of each phase depends on its own pressure

Khalil, Z., and Saad, M. Degenerate two-phase compressible immiscible flow in porous media: the case where the density of each phase depends on its own pressure. Math. Comput. Simulation 81 (2011), 2225–2233. http://dx.doi.org/10.1016/j.matcom.2010.12.012

2011 doi
[162]

A posteriori error analysis for locally conservative mixed methods

Kim, K.-Y. A posteriori error analysis for locally conservative mixed methods. Math. Comp. 76 (2007), 43–66. http://dx.doi.org/10.1090/S0025-5718-06-01903-X

2007 doi
[163]

A posteriori error estimators for locally conservative methods of nonlinear elliptic problems

Kim, K.-Y. A posteriori error estimators for locally conservative methods of nonlinear elliptic problems. Appl. Numer. Math. 57 (2007), 1065–1080. http://dx.doi.org/10.1016/j.apnum. 2006.09.010

2007 doi
[164]

A., and Russell, T

Klausen, R. A., and Russell, T. F. Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput. Geosci. 8 (2004), 341–377 (2005). http: //dx.doi.org/10.1007/s10596-005-1815-9 . 79

2004 doi
[165]

Adaptive simulations of two-phase flow by discontinuous Galerkin methods

Klieber, W., and Rivi` ere, B. Adaptive simulations of two-phase flow by discontinuous Galerkin methods. Comput. Methods Appl. Mech. Engrg. 196 (2006), 404–419. http://dx.doi.org/10. 1016/j.cma.2006.05.007

2006
[166]

Solute transport in porous media with equilibrium and nonequilibrium multiple-site adsorption: uniqueness of weak solutions

Knabner, P., and Otto, F. Solute transport in porous media with equilibrium and nonequilibrium multiple-site adsorption: uniqueness of weak solutions. Nonlinear Anal. 42 (2000), 381–403

2000
[167]

Linearly decoupled energy-stable numerical methods for multi- component two-phase compressible flow

Kou, J., Sun, S., and Wang, X. Linearly decoupled energy-stable numerical methods for multi- component two-phase compressible flow. SIAM J. Numer. Anal. 56 (2018), 3219–3248. https: //doi.org/10.1137/17M1162287

2018 doi
[168]

A posteriori error estimates and adap- tive methods for hyperbolic and convection dominated parabolic conservation laws

Kr¨ oner, D., K¨ uther, M., Ohlberger, M., and Rohde, C. A posteriori error estimates and adap- tive methods for hyperbolic and convection dominated parabolic conservation laws. In Trends in nonlinear analysis. Springer, Berlin, 2003, pp. 289–306

2003
[169]

Flow of oil and water in a porous medium

Kr¨ oner, D., and Luckhaus, S. Flow of oil and water in a porous medium. J. Differential Equations 55 (1984), 276–288. http://dx.doi.org/10.1016/0022-0396(84)90084-6

1984 doi
[170]

A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions

Kr¨ oner, D., and Ohlberger, M. A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions. Math. Comp. 69 (2000), 25–39. http://dx.doi. org/10.1090/S0025-5718-99-01158-8

2000 doi
[171]

M., and Repin, S

Kumar, K., Kyas, S., Nordbotten, J. M., and Repin, S. Guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the Biot problem. Comput. Math. Appl. 91 (2021), 122–149. https://doi.org/10.1016/j.camwa.2020.05.005

2021 doi
[172]

New mixed finite element method on polygonal and polyhedral meshes

Kuznetsov, Y., and Repin, S. New mixed finite element method on polygonal and polyhedral meshes. Russian J. Numer. Anal. Math. Modelling 18 (2003), 261–278. http://dx.doi.org/10. 1163/156939803322380846

2003
[173]

Kuznetsov, Y. A. Mixed finite element methods on polyhedral meshes for diffusion equations. In Partial differential equations , vol. 16 of Comput. Methods Appl. Sci. Springer, Dordrecht, 2008, pp. 27–41

2008
[174]

Ladev` eze, P.Comparaison de mod` eles de milieux continus. Ph.D. thesis, Universit´ e Pierre et Marie Curie (Paris 6), 1975

1975
[175]

Error estimate procedure in the finite element method and appli- cations

Ladev` eze, P., and Leguillon, D. Error estimate procedure in the finite element method and appli- cations. SIAM J. Numer. Anal. 20 (1983), 485–509

1983
[176]

Translated from the 2001 French original by Theofanis Strouboulis

Ladev` eze, P., and Pelle, J.-P. Translated from the 2001 French original by Theofanis Strouboulis. Mastering calculations in linear and nonlinear mechanics. Mechanical Engineering Series. Springer- Verlag, New York, 2005

2001
[177]

G., and Niklasson, A

Larson, M. G., and Niklasson, A. J. A conservative flux for the continuous Galerkin method based on discontinuous enrichment. Calcolo 41 (2004), 65–76. https://doi.org/10.1007/ s10092-004-0084-7

2004
[178]

A posteriori error estimates for finite volume element approximations of convection-diffusion-reaction equations

Lazarov, R., and Tomov, S. A posteriori error estimates for finite volume element approximations of convection-diffusion-reaction equations. Comput. Geosci. 6 (2002), 483–503. Locally conservative numerical methods for flow in porous media, https://doi.org/10.1023/A:1021247300362

2002 doi
[179]

T., and Wheeler, M

Li, H., Leung, W. T., and Wheeler, M. F. Sequential local mesh refinement solver with separate temporal and spatial adaptivity for non-linear two-phase flow problems. J. Comput. Phys. 403 (2020), 109074, 23. https://doi.org/10.1016/j.jcp.2019.109074

2020
[180]

A posteriori error estimates for the weak Galerkin finite element methods on polytopal meshes

Li, H., Mu, L., and Ye, X. A posteriori error estimates for the weak Galerkin finite element methods on polytopal meshes. Commun. Comput. Phys. 26 (2019), 558–578. https://doi.org/10.4208/ cicp.oa-2018-0058

2019
[181]

Li, H., and Wheeler, M. F. Dynamic local coupling for multiphase flow: a compromise between efficiency and stability. J. Comput. Phys. 469 (2022), Paper No. 111535, 14. https://doi.org/ 10.1016/j.jcp.2022.111535. 80

2022
[182]

Liu, L., and Keyes, D. E. Approximate error bounds on solutions of nonlinearly preconditioned PDEs. SIAM J. Sci. Comput. 43 (2021), A2526–A2554. https://doi.org/10.1137/19M1285044

2021 doi
[183]

Luce, R., and Wohlmuth, B. I. A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42 (2004), 1394–1414. http://dx.doi.org/10.1137/S0036142903433790

2004 doi
[184]

A., and White, Jr., A

Manteuffel, T. A., and White, Jr., A. B. The numerical solution of second-order boundary value problems on nonuniform meshes. Math. Comp. 47 (1986), 511–535, S53–S55. https://doi.org/ 10.2307/2008170

1986 doi
[185]

Construction process for an improved meshing for the simulation of a reservoir in an underground formation

Mesri, Y., and Ricois, O. Construction process for an improved meshing for the simulation of a reservoir in an underground formation. Patent CA 28886110, http://brevets-patents.ic.gc. ca/opic-cipo/cpd/fra/brevet/2886110/, 23 03, 2015

2015
[186]

A posteriori error analysis for finite element approximation of some groundwater models

Mghazli, Z. A posteriori error analysis for finite element approximation of some groundwater models. Part I: Linear case. In Error control, adaptive discretizations, and applications. Part 3 , vol. 60 of Adv. Appl. Mech. Academic Press, San Diego, CA, 2025. In press

2025
[187]

A posteriori error analysis for finite element approximation of some groundwater models

Mghazli, Z. A posteriori error analysis for finite element approximation of some groundwater models. Part II: Richards equation and mixed finite element. In Error control, adaptive discretizations, and applications. Part 4 , vol. 61 of Adv. Appl. Mech. Academic Press, San Dieg...

2025
[188]

http://dx.doi.org/10.1090/S0025-5718-2013-02723-8

2013 doi
[189]

Mitchell, W. F. A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput. 220 (2013), 350–364. https://doi.org/10.1016/j.amc.2013.05.068

2013 doi
[190]

Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms

Mitra, K., and Vohral ´ ık, M. Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on iterative linearization. HAL Preprint 04156711, submitted for publication, ...

2023
[191]

A posteriori error estimates for the Richards equation

Mitra, K., and Vohral ´ ık, M. A posteriori error estimates for the Richards equation. Math. Comp. 93 (2024), 1053–1096. https://doi.org/10.1090/mcom/3932

2024 doi
[192]

W., and S¨ uli, E

Morton, K. W., and S¨ uli, E. Finite volume methods and their analysis. IMA J. Numer. Anal. 11 (1991), 241–260. https://doi.org/10.1093/imanum/11.2.241

1991 doi
[193]

Goal-oriented error estimation based on equilibrated-flux re- construction for finite element approximations of elliptic problems

Mozolevski, I., and Prudhomme, S. Goal-oriented error estimation based on equilibrated-flux re- construction for finite element approximations of elliptic problems. Comput. Methods Appl. Mech. Engrg. 288 (2015), 127–145. https://doi.org/10.1016/j.cma.2014.09.025

2015 doi
[194]

Residual-based a posteriori error estimation for mixed virtual element methods

Munar, M., Cangiani, A., and Vel´ asquez, I. Residual-based a posteriori error estimation for mixed virtual element methods. Comput. Math. Appl. 166 (2024), 182–197. https://doi.org/10.1016/ j.camwa.2024.05.011

2024
[195]

J., and Walkington, N

Murphy, T. J., and Walkington, N. J. Control volume approximation of degenerate two-phase porous flows. SIAM J. Numer. Anal. 57 (2019), 527–546

2019
[196]

Mixed finite elements inR3

N´ ed´ elec, J.-C. Mixed finite elements inR3. Numer. Math. 35 (1980), 315–341

1980
[197]

A posteriori error estimations of some cell-centered finite volume methods

Nicaise, S. A posteriori error estimations of some cell-centered finite volume methods. SIAM J. Numer. Anal. 43 (2005), 1481–1503

2005
[198]

A posteriori error estimations of some cell centered finite volume methods for diffusion- convection-reaction problems

Nicaise, S. A posteriori error estimations of some cell centered finite volume methods for diffusion- convection-reaction problems. SIAM J. Numer. Anal. 44 (2006), 949–978

2006
[199]

H., Siebert, K

Nochetto, R. H., Siebert, K. G., and Veeser, A. Theory of adaptive finite element methods: an introduction. In Multiscale, nonlinear and adaptive approximation . Springer, Berlin, 2009, pp. 409–

2009
[200]

M., Keilegavlen, E., and Sandvin, A

Nordbotten, J. M., Keilegavlen, E., and Sandvin, A. Mass conservative domain decomposition for porous media flow. In Finite Volume Method - Powerful Means of Engineering Design , R. Petrova, Ed. IntechOpen, Rijeka, 2012, ch. 11. https://doi.org/10.5772/38700

2012 doi
[201]

M., and Keilegavlen, E

Nordbotten, J. M., and Keilegavlen, E. An introduction to multi-point flux (MPF A) and stress (MPSA) finite volume methods for thermo-poroelasticity. In Polyhedral methods in geosciences , vol. 27 of SEMA SIMAI Springer Ser. Springer, Cham, [2021] ©2021, pp. 119–158. https: //...

2021 doi
[202]

A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection–diffusion equations

Ohlberger, M. A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection–diffusion equations. Numer. Math. 87 (2001), 737–761

2001
[203]

Odeh, A. S. Comparison of solutions to a three-dimensional black-oil reservoir simulation problem (includes associated paper 9741). J. Pet. Technol. 33 (1981), 13–25. https://doi.org/10.2118/ 9723-PA

1981
[204]

Adaptive finite volume approximations for weakly coupled convection dominated parabolic systems

Ohlberger, M., and Rohde, C. Adaptive finite volume approximations for weakly coupled convection dominated parabolic systems. IMA J. Numer. Anal. 22 (2002), 253–280

2002
[205]

A posteriori error estimates for vertex centered finite volume approximations of convection–diffusion–reaction equations

Ohlberger, M. A posteriori error estimates for vertex centered finite volume approximations of convection–diffusion–reaction equations. M2AN Math. Model. Numer. Anal. 35 (2001), 355–387

2001
[206]

A posteriori error estimation for the discrete duality finite volume discretization of the Laplace equation

Omnes, P., Penel, Y., and Rosenbaum, Y. A posteriori error estimation for the discrete duality finite volume discretization of the Laplace equation. SIAM J. Numer. Anal. 47 (2009), 2782–2807

2009
[207]

A., and Tyrtyshnikov, E

Olshanskii, M. A., and Tyrtyshnikov, E. E. Theory and applications. Iterative methods for linear systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2014. http://dx. doi.org/10.1137/1.9781611973464

2014 doi
[208]

L1-contraction and uniqueness for unstationary saturated-unsaturated porous media flow

Otto, F. L1-contraction and uniqueness for unstationary saturated-unsaturated porous media flow. Adv. Math. Sci. Appl. 7 (1997), 537–553

1997
[209]

L1-contraction and uniqueness for quasilinear elliptic-parabolic equations

Otto, F. L1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differential Equations 131 (1996), 20–38. http://dx.doi.org/10.1006/jdeq.1996.0155

1996
[210]

Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach

Papeˇ z, J., R¨ ude, U., Vohral ´ ık, M., and Wohlmuth, B. Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach. Recovering mass balance in any situation. Comput. Methods Appl. Mech. Engrg. 371 (2020), 113243. https://doi.or...

2020
[211]

Algebraic error in numerical PDEs and its estimation

Papeˇ z, J. Algebraic error in numerical PDEs and its estimation. In Error control, adaptive dis- cretizations, and applications. Part 1 , vol. 58 of Adv. Appl. Mech. Academic Press, San Diego, CA, 2024, pp. 377–427

2024
[212]

E., and Weinberger, H

Payne, L. E., and Weinberger, H. F. An optimal Poincar´ e inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960), 286–292

1960
[213]

T., and Rønquist, E

Patera, A. T., and Rønquist, E. M. A general output bound result: application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci. 11 (2001), 685–712. http://dx.doi.org/10.1142/S0218202501001057

2001 doi
[214]

Adaptive finite elements for a linear parabolic problem

Picasso, M. Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998), 223–237. http://dx.doi.org/10.1016/S0045-7825(98)00121-2

1998 doi
[215]

V., Vohral ´ ık, M., Wheeler, M

Pencheva, G. V., Vohral ´ ık, M., Wheeler, M. F., and Wildey, T. Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling. SIAM J. Numer. Anal. 51 (2013), 526–554. http://dx.doi.org/10.1137/110839047

2013 doi
[216]

Prager, W., and Synge, J. L. Approximations in elasticity based on the concept of function space. Quart. Appl. Math. 5 (1947), 241–269

1947
[217]

A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements

Picasso, M. A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements. Comm. Numer. Methods Engrg. 25 (2009), 339–355. https: //doi.org/10.1002/cnm.1120

2009 doi
[218]

A mixed finite element method for 2nd order elliptic prob- lems

Raviart, P.-A., and Thomas, J.-M. A mixed finite element method for 2nd order elliptic prob- lems. In Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) . Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math.,...

1975
[219]

Adaptive finite element analysis of nonlinear problems: balancing of discretization and iteration errors

Rannacher, R., and Vihharev, J. Adaptive finite element analysis of nonlinear problems: balancing of discretization and iteration errors. J. Numer. Math. 21 (2013), 23–61. https://doi.org/10. 1515/jnum-2013-0002. 82

2013
[220]

Strict bounding of quantities of interest in computations based on domain decomposition

Rey, V., Gosselet, P., and Rey, C. Strict bounding of quantities of interest in computations based on domain decomposition. Comput. Methods Appl. Mech. Engrg. 287 (2015), 212–228. http: //dx.doi.org/10.1016/j.cma.2015.01.009

2015 doi
[221]

A posteriori estimates for partial differential equations , vol

Repin, S. A posteriori estimates for partial differential equations , vol. 4 of Radon Series on Computational and Applied Mathematics . Walter de Gruyter GmbH & Co. KG, Berlin, 2008. http://dx.doi.org/10.1515/9783110203042

2008 doi
[222]

A strict error bound with separated contributions of the discretiza- tion and of the iterative solver in non-overlapping domain decomposition methods.Comput

Rey, V., Rey, C., and Gosselet, P. A strict error bound with separated contributions of the discretiza- tion and of the iterative solver in non-overlapping domain decomposition methods.Comput. Methods Appl. Mech. Engrg. 270 (2014), 293–303. http://dx.doi.org/10.1016/j.cma.2013.12.001

2014 doi
[223]

Strict lower bounds with separation of sources of error in non-overlapping domain decomposition methods

Rey, V., Gosselet, P., and Rey, C. Strict lower bounds with separation of sources of error in non-overlapping domain decomposition methods. Internat. J. Numer. Methods Engrg. 108 (2016), 1007–1029. http://dx.doi.org/10.1002/nme.5244

2016 doi
[224]

E., and Thomas, J.-M

Roberts, J. E., and Thomas, J.-M. Mixed and hybrid methods. In Handbook of Numerical Analysis, Vol. II. North-Holland, Amsterdam, 1991, pp. 523–639

1991
[225]

Vision g´ en´ erale du simulateur ARCEOR

Ricois, O. Vision g´ en´ erale du simulateur ARCEOR. Tech. rep., IFPEN, 2011. Number 62096

2011
[226]

Mixed finite element–discontinuous finite volume element dis- cretization of a general class of multicontinuum models

Ruiz-Baier, R., and Lunati, I. Mixed finite element–discontinuous finite volume element dis- cretization of a general class of multicontinuum models. J. Comput. Phys. 322 (2016), 666–688. https://doi.org/10.1016/j.jcp.2016.06.054

2016 doi
[227]

Rose, M. E. Compact finite volume methods for the diffusion equation. J. Sci. Comput. 4 (1989), 261–290. https://doi.org/10.1007/BF01061058

1989 doi
[228]

Front tracking for two-phase flow in reservoir simulation by adaptive mesh

Saad, M., and Zhang, H. Front tracking for two-phase flow in reservoir simulation by adaptive mesh. Numer. Methods Partial Differential Equations 13 (1997), 673–697. http://dx.doi.org/ 10.1002/(SICI)1098-2426(199711)13:6<673::AID-NUM5>3.0.CO;2-O

1997 doi
[229]

F., and Wheeler, M

Russell, T. F., and Wheeler, M. F. Finite element and finite difference methods for continuous flows in porous media. In The Mathematics of Reservoir Simulation . SIAM, Philadelphia, 1983, pp. 35–106

1983
[230]

A composite mixed finite element for hexahedral grids

Sboui, A., Jaffr´ e, J., and Roberts, J. A composite mixed finite element for hexahedral grids. SIAM J. Sci. Comput. 31 (2009), 2623–2645. http://dx.doi.org/10.1137/070703703

2009 doi
[231]

Iterative methods for sparse linear systems , second ed

Saad, Y. Iterative methods for sparse linear systems , second ed. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003

2003
[232]

An introduction to the a posteriori error analysis of parabolic partial differential equa- tions

Smears, I. An introduction to the a posteriori error analysis of parabolic partial differential equa- tions. In Error control, adaptive discretizations, and applications. Part 4 , vol. 61 of Adv. Appl. Mech. Academic Press, San Diego, CA, 2025. In press

2025
[233]

Existence of solutions for a model of multiphase flow in porous media applied to gas migration in underground nuclear waste repository

Sma ¨ ı, F. Existence of solutions for a model of multiphase flow in porous media applied to gas migration in underground nuclear waste repository. Appl. Anal. 88 (2009), 1609–1616. https: //doi.org/10.1080/00036810902942226

2009 doi
[234]

A posteriori error analysis and global error control for adaptive finite volume approximations of hyperbolic problems

S¨ uli, E. A posteriori error analysis and global error control for adaptive finite volume approximations of hyperbolic problems. In Numerical analysis 1995 (Dundee, 1995) , vol. 344 of Pitman Res. Notes Math. Ser. Longman, Harlow, 1996, pp. 169–190

1995
[235]

S., Mitra, K., Storvik, E., Both, J

Stokke, J. S., Mitra, K., Storvik, E., Both, J. W., and Radu, F. A. An adaptive solution strategy for Richards’ equation. Comput. Math. Appl. 152 (2023), 155–167. https://doi.org/10.1016/ j.camwa.2023.10.020

2023
[236]

Synge, J. L. The hypercircle in mathematical physics: a method for the approximate solution of boundary value problems . Cambridge University Press, New York, 1957

1957
[237]

A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems

S¨ uli, E. A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In An introduction to recent developments in theory and numerics for conservation laws (Freiburg/Littenweiler, 1997), vol. 5 of Lect. Notes Comput. Sci. Eng. Springe...

1997 doi
[238]

Robusta posteriori error estimates for stabilized finite element meth- ods

Tobiska, L., and Verf¨ urth, R. Robusta posteriori error estimates for stabilized finite element meth- ods. IMA J. Numer. Anal. 35 (2015), 1652–1671. https://doi.org/10.1093/imanum/dru060

2015 doi
[239]

Sur l’analyse num´ erique des m´ ethodes d’´ el´ ements finis hybrides et mixtes

Thomas, J.-M. Sur l’analyse num´ erique des m´ ethodes d’´ el´ ements finis hybrides et mixtes. Ph.D. dissertation, Universit´ e Pierre et Marie Curie (Paris 6), May 1977

1977
[240]

A posteriori error estimators for convection-diffusion equations

Verf¨ urth, R. A posteriori error estimators for convection-diffusion equations. Numer. Math. 80 (1998), 641–663. https://doi.org/10.1007/s002110050381

1998 doi
[241]

van der Vorst, H. A. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 13 (1992), 631–644

1992
[242]

A posteriori error estimates for finite element discretizations of the heat equation

Verf¨ urth, R. A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40 (2003), 195–212. http://dx.doi.org/10.1007/s10092-003-0073-2

2003 doi
[243]

Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation

Verf¨ urth, R. Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math. 78 (1998), 479–493. https://doi.org/10.1007/s002110050322

1998 doi
[244]

Robust a posteriori error estimates for stationary convection-diffusion equations.SIAM J

Verf¨ urth, R. Robust a posteriori error estimates for stationary convection-diffusion equations.SIAM J. Numer. Anal. 43 (2005), 1766–1782. http://dx.doi.org/10.1137/040604261

2005 doi
[245]

Robust a posteriori error estimates for nonstationary convection-diffusion equations

Verf¨ urth, R. Robust a posteriori error estimates for nonstationary convection-diffusion equations. SIAM J. Numer. Anal. 43 (2005), 1783–1802. http://dx.doi.org/10.1137/040604273

2005 doi
[246]

Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes

Vohral ´ ık, M. Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes. M2AN Math. Model. Numer. Anal. 40 (2006), 367–391. http: //dx.doi.org/10.1051/m2an:2006013

2006 doi
[247]

A posteriori error estimation techniques for finite element methods

Verf¨ urth, R. A posteriori error estimation techniques for finite element methods . Numeri- cal Mathematics and Scientific Computation. Oxford University Press, Oxford, 2013. http: //dx.doi.org/10.1093/acprof:oso/9780199679423.001.0001

2013
[248]

A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization

Vohral ´ ık, M. A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization. C. R. Math. Acad. Sci. Paris 346 (2008), 687–690. http://dx.doi.org/10.1016/j.crma.2008.03.006

2008 doi
[249]

A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations

Vohral ´ ık, M. A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal. 45 (2007), 1570–1599. http: //dx.doi.org/10.1137/060653184

2007 doi
[250]

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohral ´ ık, M. Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. Math. Comp. 79 (2010), 2001–2032. http://dx.doi.org/10.1090/ S0025-5718-2010-02375-0

2010
[251]

Residual flux-based a posteriori error estimates for finite volume and related lo- cally conservative methods

Vohral ´ ık, M. Residual flux-based a posteriori error estimates for finite volume and related lo- cally conservative methods. Numer. Math. 111 (2008), 121–158. http://dx.doi.org/10.1007/ s00211-008-0168-4

2008
[252]

Lecture notes

Vohral ´ ık, M. Lecture notes. Analysis and approximation of partial differential equations by finite elements. ENSTA, Institut Polytechnique de Paris, 2025. https://who.rocq.inria.fr/Martin. Vohralik/Enseig/FEM/FEM.pdf

2025
[253]

Lecture notes.A posteriori error estimates for efficiency and error control in numerical simulations

Vohral ´ ık, M. Lecture notes.A posteriori error estimates for efficiency and error control in numerical simulations. Charles University, Prague, 2024. https://who.rocq.inria.fr/Martin.Vohralik/ Enseig/APost/a_posteriori.pdf

2024
[254]

Vohral ´ ık, M., and Wohlmuth, B. I. Mixed finite element methods: implementation with one un- known per element, local flux expressions, positivity, polygonal meshes, and relations to other methods. Math. Models Methods Appl. Sci. 23 (2013), 803–838. https://doi.org/10.1142/ ...

2013
[255]

Vohral ´ ık, M., and Wheeler, M. F. A posteriori error estimates, stopping criteria, and adaptiv- ity for two-phase flows. Comput. Geosci. 17 (2013), 789–812. http://dx.doi.org/10.1007/ s10596-013-9356-0 . 84

2013
[256]

A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra

Wheeler, M., Xue, G., and Yotov, I. A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numer. Math. 121 (2012), 165–204. http://dx.doi.org/10.1007/ s00211-011-0427-7

2012
[257]

A simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows

Vohral ´ ık, M., and Yousef, S. A simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows. Comput. Methods Appl. Mech. Engrg. 331 (2018), 728–760. https://doi.org/10.1016/j.cma.2017.11.027

2018 doi
[258]

Geometric integration theory

Whitney, H. Geometric integration theory. Princeton University Press, Princeton, NJ, 1957

1957
[259]

F., and Yotov, I

Wheeler, M. F., and Yotov, I. A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44 (2006), 2082–2106. http://dx.doi.org/10.1137/050638473

2006 doi
[260]

A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements

Youn` es, A., Mos´ e, R., Ackerer, P., and Chavent, G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. J. Comput. Phys. 149 (1999), 148–167

1999
[261]

From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions

Youn` es, A., Ackerer, P., and Chavent, G. From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions. Internat. J. Numer. Methods Engrg. 59 (2004), 365–388

2004
[262]

A posteriori error estimates and adaptivity based on stopping criteria and adaptive mesh refinement for multiphase and thermal flows

Yousef, S. A posteriori error estimates and adaptivity based on stopping criteria and adaptive mesh refinement for multiphase and thermal flows. Application to steam-assisted gravity drainage . Ph.D. thesis, Universit´ e Pierre et Marie Curie - Paris VI, 2013. https://tel.arch...

2013
[263]

C., and Stephenson, R

Young, L. C., and Stephenson, R. E. A generalized compositional approach for reservoir simula- tion. Society of Petroleum Engineers Journal 23 (1983), 727–742. https://doi.org/10.2118/ 10516-PA

1983
[265]

Nonlinear functional analysis and its applications

Zeidler, E. Nonlinear functional analysis and its applications. II/B . Springer-Verlag, New York,
[542]

http://dx.doi.org/10.1007/978-3-642-03413-8_12

doi
[1990]

https://doi.org/10.1007/978-1-4612-0985-0 . 85

doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Discretization on unstructured grids for inhomogeneous, anisotropic media

Aavatsmark, I., Barkve, T., Bøe, Ø., and Mannseth, T. Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19 (1998), 1700–1716

1998

[2] [2]

T., Klausen, R

Aavatsmark, I., Eigestad, G. T., Klausen, R. A., Wheeler, M. F., and Yotov, I. Convergence of a symmetric MPF A method on quadrilateral grids. Comput. Geosci. 11 (2007), 333–345. http: //dx.doi.org/10.1007/s10596-007-9056-8

work page doi:10.1007/s10596-007-9056-8 2007

[3] [3]

A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations

Achdou, Y., Bernardi, C., and Coquel, F. A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations. Numer. Math. 96 (2003), 17–42. http://dx.doi.org/10. 1007/s00211-002-0436-7

2003

[4] [4]

General purpose compositional model

´Acs, G., Doleschall, S., and Farkas, ´E. General purpose compositional model. Society of Petroleum Engineers Journal 25 (1985), 543–553. https://doi.org/10.2118/10515-PA

doi:10.2118/10515-pa 1985

[5] [5]

Adams, R. A. Pure and Applied Mathematics, Vol. 65. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975

1975

[6] [6]

A posteriori estimators for the finite vol- ume discretization of an elliptic problem

Afif, M., Bergam, A., Mghazli, Z., and Verf¨ urth, R. A posteriori estimators for the finite vol- ume discretization of an elliptic problem. Numer. Algorithms 34 (2003), 127–136. International Conference on Numerical Algorithms, Vol. II (Marrakesh, 2001)

2003

[7] [7]

Connection between finite volume and mixed finite element methods for a diffusion problem with nonconstant coefficients

Agouzal, A., Baranger, J., Ma ˆ ıtre, J.-F., and Oudin, F. Connection between finite volume and mixed finite element methods for a diffusion problem with nonconstant coefficients. Application to a convection diffusion problem. East-West J. Numer. Math. 3 (1995), 237–254

1995

[8] [8]

A posteriori error estimator for finite volume methods

Agouzal, A., and Oudin, F. A posteriori error estimator for finite volume methods. Appl. Math. Comput. 110 (2000), 239–250

2000