Pith. sign in

REVIEW 3 major objections 6 minor 51 references

Subspace and auxiliary space preconditioners for high-order interior penalty discretizations in $H(\mathrm{div})$

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Vertex-patch, fictitious-space, and auxiliary-space preconditioners for H(div) DG discretizations yield condition numbers independent of mesh size and, on affine meshes, of polynomial degree and penalty parameter.

desk verdict A solid, useful preconditioning paper for H(div)-conforming IPDG with convincing numerics and a few repairable proof gaps, most notably the coarse-space counting in Lemma 9. read the letter →

arxiv 2411.14561 v1 pith:KIO3XKWP submitted 2024-11-21 math.NA cs.NA

Will Pazner This is my paper · ORCID
classification math.NAcs.NA MSC 65N3065N2265F0865N55
keywords H(div)finiteelementsinteriorpenaltydiscontinuousGalerkinsubspacecorrectionpreconditionerfictitiousspaceauxiliaryvertexpatchGauss-LobattointerpolationStokessaddle-pointpreconditioning
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 builds and analyzes preconditioners for the interior penalty discontinuous Galerkin discretization of the vector Poisson problem posed in $H(\mathrm{div})$, the finite element space of functions with continuous normal components. Such discretizations are the velocity block in exactly divergence-free, pressure-robust Stokes solvers, so a good preconditioner there unlocks scalable incompressible-flow simulation. The paper claims three preconditioners—a subspace correction method using vertex patches and a lowest-order $H^1$-conforming coarse space, a fictitious space method using the degree-$p$ discontinuous Galerkin space, and an auxiliary space method using the degree-$(p-1)$ DG space with a block Jacobi smoother—produce preconditioned systems whose condition numbers are independent of mesh size $h$; on affine (parallelogram) meshes, the subspace and fictitious space methods are additionally independent of polynomial degree $p$ and penalty parameter $\eta$. Numerical experiments on structured and unstructured meshes show iteration counts that grow only mildly with $p$ even when the theory does not guarantee $p$-independence, and the methods drive uniform MINRES convergence when used as the velocity block of a block-diagonal Stokes preconditioner.

What carries the argument

The load-bearing object is the subspace decomposition $V_h = V_0 + \sum_i V_i$, where $V_0$ is the lowest-order $H^1$-conforming space and $V_i$ are vertex-patch spaces of Raviart-Thomas functions supported on the union of elements around each vertex. The argument hinges on two inequalities: a finite-overlap bound that limits the cross terms among the $V_i$ in the $a(\cdot,\cdot)$ inner product, and a stable-decomposition estimate that builds an explicit split of any $v_h$ into a coarse part (via Oswald averaging and vertex interpolation) and local parts (via Gauss-Lobatto nodal interpolation of vertex-patch products), with all constants controlled by the explicit Gauss-Lobatto interpolation stability lemmas in Section 4. The fictitious space and auxiliary space methods reuse the same interpolation and averaging machinery through the transfer operator $R = \hat{Q} I_{\hat{V}_h}$, where $I_{\hat{V}_h}$ is nodal interpolation into the broken $H(\mathrm{div})$ space and $\hat{Q}$ is a diagonal averaging operator, reducing the $H(\mathrm{div})$ problem to a standard DG problem whose inverse can be approximated by spectrally equivalent low-order refined preconditioners.

What would settle it

Run the subspace-correction preconditioner on a family of meshes formed by subdividing a disk into $N$ sectors meeting at a single central vertex, so that the central vertex's valence grows with $N$, and plot the condition number of $BA$ against $N$; a condition number that grows with $N$ (or with the total vertex count on any fixed-valence refinement) would falsify the claimed $h$-independence.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1: for the subspace correction preconditioner $B$, the condition number of the preconditioned operator $BA$ is bounded independently of the mesh size $h$, and when all mesh elements are affinely mapped, independently of the polynomial degree $p$ and the penalty parameter $\eta$. Theorems 2 and 3 extend the $h$-independence to a fictitious space preconditioner built from the degree-$p$ discontinuous Galerkin space and to an auxiliary space preconditioner built from the degree-$(p-1)$ DG space with a block Jacobi smoother, with $p$- and $\eta$-independence shown on Cartesian (or affinely transformed Cartesian) meshes. The analysis goes through a stable subspace decomposition into vertex patches plus a coarse space, with the stability constants controlled by explicit Gauss-Lobatto interpolation estimates, and through a transfer operator between the $H(\mathrm{div})$ space and the piecewise polynomial DG space. On general non-affine meshes the paper proves only $h$-independence, but reports numerical evidence that $p$-dependence stays mild.

Load-bearing premise

The $h$-independence bound depends on the subspace overlap counting staying bounded by a constant, but the coarse space $V_0$ overlaps with every vertex patch, and the paper provides no separate estimate for those coarse-to-patch cross terms.

Editorial extensions

If this is right

  • If the $h$-independence holds as stated, the subspace correction preconditioner becomes a drop-in replacement for the velocity-block inverse in block-diagonal Stokes preconditioners, making the full saddle-point solve uniform in mesh size without global iterative tuning.
  • The fictitious space preconditioner, being expressible through Kronecker-product sum factorization and a low-order refined DG operator, can be implemented matrix-free with $O(p^{d+1})$ work and $O(p^d)$ memory per element, so high-order $H(\mathrm{div})$ discretizations become practical at very high $p$.
  • The auxiliary space preconditioner, which avoids the expensive vertex-patch solves, offers a cheaper route to $p$-robust convergence on unstructured meshes, and the paper's numerical results indicate it is the fastest of the three at high order on non-affine meshes.
  • On affine meshes, independence from the penalty parameter $\eta$ means users are free to choose $\eta$ for stability or accuracy without worrying about solver degradation, which the paper notes is especially relevant for the $\eta$-sensitive inf-sup constant in Stokes.

Reading between the lines

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

  • A strict reading of Lemma 9 suggests the proof's overlap count treats the coarse space $V_0$ as a single additional overlap, but $V_0$ interacts with every vertex patch; establishing the full $h$-independence on general meshes would need a separate estimate for coarse-to-patch cross terms.
  • The explicit Gauss-Lobatto constants in Section 4 give the unproved bound $\kappa \lesssim \eta p^2$ for the subspace preconditioner on non-affine meshes, so explaining the observed mild numerical growth would require either sharper trace estimates or a different vertex-patch decomposition.
  • The fictitious space construction naturally suggests a full $p$-multigrid cycle by recursively applying the degree-$(p-1)$ auxiliary space idea, an extension the paper does not pursue.
  • Because the low-order refined operator used in the fictitious space preconditioner is an M-matrix written as a weighted graph Laplacian, classical algebraic multigrid theory could be invoked to prove uniform convergence of the approximate solver, not just the ideal inverse, on general meshes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper develops and analyzes three preconditioners for the interior penalty discontinuous Galerkin discretization of a vector Poisson problem posed in the H(div)-conforming Raviart-Thomas space: an additive subspace correction preconditioner using vertex patches and a lowest-order H1-conforming coarse space, a fictitious space preconditioner using the degree-p discontinuous Galerkin space, and an auxiliary space preconditioner using the degree-(p-1) discontinuous Galerkin space with a block Jacobi smoother. The main theoretical claims are that the subspace correction preconditioner has a condition number independent of h, and also independent of p and eta on affine meshes (Theorem 1); the fictitious space preconditioner is h-robust in general and p/eta-robust on Cartesian meshes (Theorem 2); and the auxiliary space preconditioner is h-robust on general meshes (Theorem 3). Section 4 gives explicit Gauss-Lobatto interpolation stability and error estimates with sharp constants. Numerical experiments on Cartesian, affine unstructured, and non-affine skewed meshes, together with block-diagonal preconditioning of an exactly divergence-free Stokes discretization, support the predicted robustness and show mild p-dependence in the cases without full theoretical guarantees.

Significance. If the theoretical claims are fully established, the paper makes a useful contribution to scalable solvers for H(div)-conforming DG methods, which are important for pressure-robust, exactly divergence-free discretizations of incompressible flow. The matrix-free implementation of the fictitious space preconditioner using low-order-refined spectral equivalence is practically relevant, and the explicit Gauss-Lobatto interpolation estimates in Section 4 are a useful technical contribution. The numerical study is reasonably broad, covering parameter sweeps in h, p, and eta on three mesh families as well as Stokes/MINRES. The main weaknesses are several proof gaps in the general-mesh parts of Theorems 1-3; these concern the coarse-space interaction in Lemma 9, the H1-stability step applied to a non-H1 function in Lemma 10, and the interpolation estimates for non-polynomial functions in the proof of Theorem 3. The affine and Cartesian cases are better supported by the text.

major comments (3)
  1. [Section 3.1, Lemma 9] Lemma 9 (Section 3.1, Eq. (14)): the lower spectral bound is proved by counting non-negligible cross terms a(vi,vj) and claiming that the count is bounded by the mesh valence plus one for the coarse space V0. This is only valid for pairs of vertex-patch spaces. The global coarse space V0 has non-negligible a-inner products with every vertex-patch space Vi, so a termwise application of Cauchy-Schwarz and Young introduces a factor depending on the number of vertices K. The resulting lower bound is therefore not h-independent as written. The gap is repairable by grouping all local components into v_rest = sum_{i>=1} vi, applying Young once to a(v0, v_rest), and then using finite overlap only among the local spaces, but this argument is absent from the manuscript. As written, the proof does not establish the h-independent lower bound that Theorem 1 requires.
  2. [Section 3.1, Lemma 10] Lemma 10 (Section 3.1): in the general non-affine case, the proof bounds the coarse term by asserting H1 stability of the nodal interpolant IV0 applied to Qh(vh). However, Section 2.2 explicitly states that Qh(vh) belongs to [H1]^d only when all mesh elements are affine; on non-affine meshes this is not generally true. Thus the argument that |||v0|||^2 is controlled by |||vh|||^2 is not justified for the general meshes covered by the h-independence part of Theorem 1. The affine case is fine, but the general-mesh statement needs a separate argument, for example one that estimates the interpolation of the possibly discontinuous averaged field directly through its jumps.
  3. [Section 3.3, proof of Theorem 3] The proof of the c0 bound in Theorem 3 sets v0 = I_W0(v) and invokes Lemma 6 and a scaling argument to obtain both stability of I_W0 on v and the estimate ||v - v0||_0^2 ≲ (h^2/p^2)||∇_h v||_0^2. On non-affine elements, a function v in Vh is not a polynomial; it is the Piola image of a polynomial under a spatially varying Jacobian, and the interpolation operator I_W0 into Q_{p-1}^d is not Piola-covariant. The cited one-dimensional polynomial interpolation results therefore do not apply directly. Since Theorem 3 only claims h-independence, constants depending on p and eta would be acceptable, but the estimate must be proved. As written, the h-independence of c0 on general meshes is not established.
minor comments (6)
  1. [Section 3.1, Lemma 10] In Eq. (16) and the surrounding text, the symbol vj appears where vh is clearly intended; this makes the proof harder to follow.
  2. [Section 3.2, proof of Theorem 2] The right-inverse S is introduced only in passing, and the phrase Svj appears to be a typo for Svh. Please define S explicitly and state how the scaling argument is uniform in h on non-affine elements.
  3. [Lemma 3] The statement says For all v in V(kappa) but the displayed inequalities use u; please align the notation.
  4. [Section 4, proof of Lemma 15] In the final display of the proof, the interpolation operator is denoted J_{p-1} instead of I_{p-1}; this is a typo.
  5. [Section 5.3, Figure 1 caption] The caption contains the duplicated phrase with with non-affine; it should read with non-affine elements.
  6. [Remark 3] The phrase On meshes with non-affine meshes is redundant; change it to On non-affine meshes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proofs reduce to independent prior domain-decomposition and fictitious/auxiliary-space theory plus self-contained polynomial estimates, not to the target results.

full rationale

The condition-number bounds in Theorems 1-3 are obtained from external abstract frameworks: Lemma 8 (Xu and Zikatanov), Lemma 11 (Nepomnyaschikh), and Lemma 14 (Xu). The finite-overlap/counting argument in Lemma 9 and the stable decomposition in Lemma 10 are worked out in the paper using auxiliary estimates (Lemmas 2-6) that are either proven in Sections 2-4 or cited from independent prior literature on hp-DG analysis. The Gauss-Lobatto interpolation estimates (Lemmas 15-17) are proved in full in Section 4 with explicit constants, so they are not assumed from the conclusion. The self-citations [34,35,36,37] are used as building blocks rather than as proof of the present theorems: [35] supplies a known L2/energy estimate, while [34,36] supply low-order-refined spectral equivalence and subspace-correction technology that are used in the implementation, not as the statement being proved. The numerical experiments are genuinely predictive, with no parameter fitted from the data used to claim uniform conditioning. The skeptical concern about Lemma 9's cross-term counting involving the coarse space V0 is a proof-completeness or correctness issue about the constant's dependence on the number of vertices K; it is not circularity, because the text does not define or fit the claimed bound in terms of itself. No fitted input is renamed as a prediction, and no known result is merely renamed. Therefore the paper shows no significant circularity.

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

The analysis rests on established finite element theory and Gauss-Lobatto interpolation results; no free parameters are fitted. The main limitations are inherited from the mesh assumptions: p- and eta-independent bounds require affine or Cartesian elements, and general meshes only give h-robustness. The paper's own Remarks 3 and 5 acknowledge that the proven p and eta bounds are pessimistic on non-affine meshes.

assumptions (6)
  • standard math Interior penalty bilinear form is continuous and coercive in the mesh-dependent DG norm (Lemma 1, citing Antonietti-Houston and Pazner-Kolev).
    Unproved in this paper; cited as Lemma 1 from [4, Lemma 2.4] and [35, Lemma 3].
  • standard math Gauss-Lobatto interpolation L2 and H1 stability constants and error estimates as proven in Section 4.
    Proved in Section 4 with explicit constants; these are standard polynomial interpolation results used throughout the preconditioner analysis.
  • domain assumption The mesh is shape-regular and consists of straight-sided quadrilaterals; p- and eta-independent bounds require affine or Cartesian elements.
    The trace inequalities in Lemma 3 and the Piola transformation arguments require bounded Jacobian condition numbers; non-affine elements are excluded from the strong bounds.
  • domain assumption The penalty parameter eta is chosen sufficiently large to guarantee coercivity of the IPDG form.
    The discrete problem and all preconditioner analyses assume the form is positive definite, which requires eta large enough.
  • domain assumption The lowest-order H1-conforming space V0 is a subspace of the H(div) space, which holds for straight-sided elements and p >= 2.
    Proved in Lemma 7 but requires p >= 2 and straight-sided elements; this inclusion is used to define the coarse space in Section 3.1.
  • domain assumption For affine elements, Piola-mapped Raviart-Thomas functions are polynomials, so the fictitious space Wh contains Vh.
    Proven in Lemma 13; this is the reason the fictitious space constant cS equals 1 on affine meshes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Subspace and auxiliary space preconditioners for high-order interior penalty discretizations in $H(\mathrm{div})$." pith.science (2026). https://pith.science/paper/KIO3XKWP

@misc{pith2026241114561,
  author       = {Pith},
  title        = {Pith review of: Subspace and auxiliary space preconditioners for high-order interior penalty discretizations in $H(\mathrmdiv)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KIO3XKWP}},
  note         = {Machine review of arXiv:2411.14561}
}
abstract

In this paper, we construct and analyze preconditioners for the interior penalty discontinuous Galerkin discretization posed in the space $H(\mathrm{div})$. These discretizations are used as one component in exactly divergence-free pressure-robust discretizations for the Stokes problem. Three preconditioners are presently considered: a subspace correction preconditioner using vertex patches and the lowest-order $H^1$-conforming space as a coarse space, a fictitious space preconditioner using the degree-$p$ discontinuous Galerkin space, and an auxiliary space preconditioner using the degree-$(p-1)$ discontinuous Galerkin space and a block Jacobi smoother. On certain classes of meshes, the subspace and fictitious space preconditioners result in provably well-conditioned systems, independent of the mesh size $h$, polynomial degree $p$, and penalty parameter $\eta$. All three preconditioners are shown to be robust with respect to $h$ on general meshes, and numerical results indicate that the iteration counts grow only mildly with respect to $p$ in the general case. Numerical examples illustrate the convergence properties of the preconditioners applied to structured and unstructured meshes. These solvers are used to construct block-diagonal preconditioners for the Stokes problem, which result in uniform convergence when used with MINRES.

Figures

Figures reproduced from arXiv: 2411.14561 by the authors.

Figure 1
Figure 1. Panels (a) and (b): unstructured mesh of affine elements (parallelograms) and uniform refinement levels ℓ used for the test case in Section 5.2. Panel (c): unstructured with with non-affine (skewed) elements used for the test case in Section 5.3. . behavior of all three preconditioners is qualitatively similar. For higher-order problems, the fictitious space and auxiliary space preconditioners display faster runtime… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 27 canonical work pages

  1. [1]

    MFEM: a modular finite element methods library

    R. Anderson et al. “MFEM: a modular finite element methods library ”. In: Computers & Mathematics with Applications (2020). doi: 10.1016/j.camwa.2020.06.009

    doi:10.1016/j.camwa.2020.06.009 2020

  2. [2]

    High-performance finite elements with MFEM

    J. Andrej et al. “High-performance finite elements with MFEM”. I n: The International Journal of High Performance Computing Applications (2024). doi: 10.1177/10943420241261981

    doi:10.1177/10943420241261981 2024

  3. [3]

    Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains

    P. F. Antonietti, S. Giani, and P. Houston. “Domain decomposition preconditioners for discontinuous Galerkin methods for elliptic problems on complicated domains”. In: Journal of Scientific Computing 60.1 (2013), pp. 203–227. doi: 10.1007/s10915-013-9792-y

    doi:10.1007/s10915-013-9792-y 2013

  4. [4]

    A class of domain decompositio n preconditioners forhp-discontinuous Galerkin finite element methods

    P. F. Antonietti and P. Houston. “A class of domain decompositio n preconditioners forhp-discontinuous Galerkin finite element methods”. In: Journal of Scientific Computing 46.1 (2010), pp. 124–149. doi: 10.1007/s10915-010-9390-1

    work page doi:10.1007/s10915-010-9390-1 2010

  5. [5]

    Algebraic multigrid schemes for hig h-order nodal discontinuous Galerkin methods

    P. F. Antonietti and L. Melas. “Algebraic multigrid schemes for hig h-order nodal discontinuous Galerkin methods”. In: SIAM Journal on Scientific Computing 42.2 (2020), A1147–A1173. doi: 10.1137/18m1204383

    work page doi:10.1137/18m1204383 2020

  6. [6]

    A uniform additive Schwarz preconditione r for high-order discontinuous Galerkin approximations of elliptic problems

    P. F. Antonietti et al. “A uniform additive Schwarz preconditione r for high-order discontinuous Galerkin approximations of elliptic problems”. In: Journal of Scientific Computing 70.2 (2016), pp. 608–630. doi: 10.1007/s10915-016-0259-9

    doi:10.1007/s10915-016-0259-9 2016

  7. [7]

    An interior penalty finite element method with discon tinuous elements

    D. N. Arnold. “An interior penalty finite element method with discon tinuous elements”. In: SIAM Journal on Numerical Analysis 19.4 (1982), pp. 742–760. doi: 10.1137/0719052. 22 REFERENCES Table 5. MINRES iteration counts (with relative tolerance of 10 −12) for the Stokes problem. The approximate block diagonal preconditioner ~B is used, where the (1 , 1)-...

    doi:10.1137/0719052 1982

  8. [8]

    Quadrilateral H(div) Finite Elements

    D. N. Arnold, D. Boffi, and R. S. Falk. “Quadrilateral H(div) Finite Elements”. In: SIAM Journal on Numerical Analysis 42.6 (2005), pp. 2429–2451. doi: 10.1137/s0036142903431924

    doi:10.1137/s0036142903431924 2005

Show all 51 references
  1. [9]

    Unified analysis of discontinuous Galerkin metho ds for elliptic problems

    D. N. Arnold et al. “Unified analysis of discontinuous Galerkin metho ds for elliptic problems”. In: SIAM Journal on Numerical Analysis 39.5 (2002), pp. 1749–1779. doi: 10.1137/S0036142901384162

    2002 doi

  2. [10]

    Matrix-free multigrid block-preconditioners f or higher order discontinuous Galerkin discretisations

    P. Bastian et al. “Matrix-free multigrid block-preconditioners f or higher order discontinuous Galerkin discretisations”. In: Journal of Computational Physics 394 (2019), pp. 417–439. doi: 10.1016/j.jcp.2019.06.001

    2019 doi

  3. [11]

    Numerical solution of sadd le point problems

    M. Benzi, G. H. Golub, and J. Liesen. “Numerical solution of sadd le point problems”. In: Acta Numerica 14 (2005), pp. 1–137. doi: 10.1017/s0962492904000212

    2005 doi

  4. [12]

    Polynomial interpolation results in S obolev spaces

    C. Bernardi and Y. Maday. “Polynomial interpolation results in S obolev spaces”. In: Journal of Com- putational and Applied Mathematics 43.1-2 (1992), pp. 53–80. doi: 10.1016/0377-0427(92)90259-z

    1992 doi

  5. [13]

    Multilevel preconditioning of discontinuous Galerkin spectral element methods. Part I: geometrically conforming meshes

    K. Brix et al. “Multilevel preconditioning of discontinuous Galerkin spectral element methods. Part I: geometrically conforming meshes”. In: IMA Journal of Numerical Analysis 35.4 (2014), pp. 1487–1532. doi: 10.1093/imanum/dru053

    2014 doi

  6. [14]

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

    E. Burman and A. Ern. “Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations”. In: Mathematics of Computation 76.259 (2007), pp. 1119–1141. doi: 10.1090/s0025-5718-07-01951-5

    2007 doi

  7. [15]

    A note on disc ontinuous Galerkin divergence-free solu- tions of the Navier–Stokes equations

    B. Cockburn, G. Kanschat, and D. Sch¨ otzau. “A note on disc ontinuous Galerkin divergence-free solu- tions of the Navier–Stokes equations”. In: Journal of Scientific Computing 31.1ˆ a2 (2006), pp. 61–73. doi: 10.1007/s10915-006-9107-7 . REFERENCES 23

    2006 doi

  8. [16]

    Multigrid methods for the Stokes pro blem on GPU systems

    C. Cui and G. Kanschat. “Multigrid methods for the Stokes pro blem on GPU systems”. arXiv eprint 2410.09497. 2024

    2024 arXiv

  9. [17]

    Two-level preconditioning of discontinuous Galerkin approximations of second- order elliptic equations

    V. A. Dobrev et al. “Two-level preconditioning of discontinuous Galerkin approximations of second- order elliptic equations”. In: Numerical Linear Algebra with Applications 13.9 (2006), pp. 753–770. doi: 10.1002/nla.504

    2006 doi

  10. [18]

    An explicit divergence-free DG method for incompressib le flow

    G. Fu. “An explicit divergence-free DG method for incompressib le flow”. In: Computer Methods in Applied Mechanics and Engineering 345 (2019), pp. 502–517. doi: 10.1016/j.cma.2018.11.012

    2019 doi

  11. [19]

    A multilevel discontinuous Galerkin method

    J. Gopalakrishnan and G. Kanschat. “A multilevel discontinuous Galerkin method”. In: Numerische Mathematik 95.3 (2003), pp. 527–550. doi: 10.1007/s002110200392

    2003 doi

  12. [20]

    Application of p-multigrid to discontinuous Galerkin formulations of the Poisson equation

    B. T. Helenbrook and H. L. Atkins. “Application of p-multigrid to discontinuous Galerkin formulations of the Poisson equation”. In: AIAA Journal 44.3 (2006), pp. 566–575. doi: 10.2514/1.15497

    2006 doi

  13. [21]

    BoomerAMG: a parallel algebraic m ultigrid solver and preconditioner

    V. E. Henson and U. M. Yang. “BoomerAMG: a parallel algebraic m ultigrid solver and preconditioner”. In: Applied Numerical Mathematics 41.1 (2002), pp. 155–177. doi: 10.1016/s0168-9274(01)00115-5

    2002 doi

  14. [22]

    On the divergence constraint in mixed finite elemen t methods for incompressible flows

    V. John et al. “On the divergence constraint in mixed finite elemen t methods for incompressible flows”. In: SIAM Review 59.3 (2017), pp. 492–544. doi: 10.1137/15m1047696

    2017 doi

  15. [23]

    Multilevel methods for discontinuous Galerkin FE M on locally refined meshes

    G. Kanschat. “Multilevel methods for discontinuous Galerkin FE M on locally refined meshes”. In: Computers & Structures 82.28 (2004), pp. 2437–2445. doi: 10.1016/j.compstruc.2004.04.015

    2004 doi

  16. [24]

    Multigrid methods for H div-conforming discontinuous Galerkin methods for the Stokes equations

    G. Kanschat and Y. Mao. “Multigrid methods for H div-conforming discontinuous Galerkin methods for the Stokes equations”. In: Journal of Numerical Mathematics 23.1 (2015). doi: 10.1515/jnma-2015-0005

    2015 doi

  17. [25]

    Conservative and accurate solution t ransfer between high-order and low- order refined finite element spaces

    T. Kolev and W. Pazner. “Conservative and accurate solution t ransfer between high-order and low- order refined finite element spaces”. In: SIAM Journal on Scientific Computing 44.1 (2022), A1–A27. doi: 10.1137/21m1403916

    2022 doi

  18. [26]

    Fast matrix-free evaluation o f discontinuous Galerkin finite element operators

    M. Kronbichler and K. Kormann. “Fast matrix-free evaluation o f discontinuous Galerkin finite element operators”. In: ACM Transactions on Mathematical Software 45.3 (2019), pp. 1–40. doi: 10.1145/3325864

    2019 doi

  19. [27]

    Fully discrete hp-finite elements: fast quadrature

    J. Melenk, K. Gerdes, and C. Schwab. “Fully discrete hp-finite elements: fast quadrature”. In: Computer Methods in Applied Mechanics and Engineering 190.32-33 (2001), pp. 4339–4364. doi: 10.1016/s0045-7825(00)00322-

    2001 doi

  20. [28]

    Decomposition and fictitious domains meth ods for elliptic boundary value prob- lems

    S. Nepomnyaschikh. “Decomposition and fictitious domains meth ods for elliptic boundary value prob- lems”. In: Proceedings of the Fifth International Conference on Domai n Decomposition Methods . Ed. by D. Keyes et al. Norfolk, Virginia, 1991

    1991

  21. [29]

    Domain decomposition methods

    S. Nepomnyaschikh. “Domain decomposition methods”. In: Lectures on Advanced Computational Meth- ods in Mechanics . Ed. by J. Kraus and U. Langer. de Gruyter, 2007

    2007

  22. [30]

    Spectral methods for problems in complex geom etries

    S. A. Orszag. “Spectral methods for problems in complex geom etries”. In: Journal of Computational Physics 37.1 (1980), pp. 70–92. doi: 10.1016/0021-9991(80)90005-4

    1980 doi

  23. [31]

    On a BPX-preconditioner for P1 elements

    P. Oswald. “On a BPX-preconditioner for P1 elements”. In: Computing 51.2 (1993), pp. 125–133. doi: 10.1007/bf02243847

    1993 doi

  24. [32]

    Additive Schwarz methods for the p-version finite element method

    L. F. Pavarino. “Additive Schwarz methods for the p-version finite element method”. In: Numerische Mathematik 66.1 (1993), pp. 493–515. doi: 10.1007/bf01385709

    1993 doi

  25. [33]

    Domain decomposition algorithms for the p-version finite element method for elliptic problems

    L. F. Pavarino. “Domain decomposition algorithms for the p-version finite element method for elliptic problems”. PhD thesis. New York University, 1992

    1992

  26. [34]

    Efficient low-order refined preconditioners for hig h-order matrix-free continuous and dis- continuous Galerkin methods

    W. Pazner. “Efficient low-order refined preconditioners for hig h-order matrix-free continuous and dis- continuous Galerkin methods”. In: SIAM Journal on Scientific Computing 42.5 (2020), A3055–A3083. doi: 10.1137/19m1282052

    2020 doi

  27. [35]

    Uniform subspace correction precon ditioners for discontinuous Galerkin methods with hp-refinement

    W. Pazner and T. Kolev. “Uniform subspace correction precon ditioners for discontinuous Galerkin methods with hp-refinement”. In: Communications on Applied Mathematics and Computation (2021). doi: 10.1007/s42967-021-00136-3

    2021 doi

  28. [36]

    Low-order precon ditioning for the high-order finite element de Rham complex

    W. Pazner, T. Kolev, and C. R. Dohrmann. “Low-order precon ditioning for the high-order finite element de Rham complex”. In: SIAM Journal on Scientific Computing 45.2 (2023), A675–A702. doi: 10.1137/22M1486534

    2023 doi

  29. [37]

    Matrix-free high-p erformance saddle-point solvers for high-order problems in H(div)

    W. Pazner, T. Kolev, and P. S. Vassilevski. “Matrix-free high-p erformance saddle-point solvers for high-order problems in H(div)”. In: SIAM Journal on Scientific Computing 46.3 (2024), B179–B204. doi: 10.1137/23m1568806

    2024 doi

  30. [38]

    Approximate tensor-product preconditioners for very high order dis- continuous Galerkin methods

    W. Pazner and P.-O. Persson. “Approximate tensor-product preconditioners for very high order dis- continuous Galerkin methods”. In: Journal of Computational Physics 354 (2018), pp. 344–369. doi: 10.1016/j.jcp.2017.10.030. 24 REFERENCES

    2018 doi

  31. [39]

    Algebraic Multigrid

    J. W. Ruge and K. St¨ uben. “Algebraic Multigrid”. In: Multigrid Methods. SIAM, 1987. Chap. 4, pp. 73–

    1987

  32. [40]

    An explicit expression for the penalty parameter of the interior penalty method

    K. Shahbazi. “An explicit expression for the penalty parameter of the interior penalty method”. In: Journal of Computational Physics 205.2 (2005), pp. 401–407. doi: 10.1016/j.jcp.2004.11.017

    2005 doi

  33. [41]

    Comparison of multigrid alg orithms for high-order continuous finite element discretizations

    H. Sundar, G. Stadler, and G. Biros. “Comparison of multigrid alg orithms for high-order continuous finite element discretizations”. In: Numerical Linear Algebra with Applications 22.4 (2015), pp. 664–

    2015

  34. [42]

    Short note on the mass matrix for Gauss–Lo batto grid points

    S. A. Teukolsky. “Short note on the mass matrix for Gauss–Lo batto grid points”. In: Journal of Computational Physics 283 (2015), pp. 408–413. doi: 10.1016/j.jcp.2014.12.012

    2015 doi

  35. [43]

    Toselli and O

    A. Toselli and O. B. Widlund. Domain Decomposition Methods — Algorithms and Theory . Springer Berlin Heidelberg, 2005. doi: 10.1007/b137868

    2005 doi

  36. [44]

    A robust numerical method for S tokes equations based on divergence- freehH(div) finite element methods

    J. Wang, Y. Wang, and X. Ye. “A robust numerical method for S tokes equations based on divergence- freehH(div) finite element methods”. In: SIAM Journal on Scientific Computing 31.4 (2009), pp. 2784–

    2009

  37. [45]

    The auxiliary space method and optimal multigrid precond itioning techniques for unstructured grids

    J. Xu. “The auxiliary space method and optimal multigrid precond itioning techniques for unstructured grids”. In: Computing 56.3 (1996), pp. 215–235. doi: 10.1007/bf02238513

    1996 doi

  38. [46]

    Iterative methods by space decomposition and subspa ce correction

    J. Xu. “Iterative methods by space decomposition and subspa ce correction”. In: SIAM Review 34.4 (1992), pp. 581–613. doi: 10.1137/1034116

    1992 doi

  39. [47]

    The method of subspace corrections

    J. Xu. “The method of subspace corrections”. In: Journal of Computational and Applied Mathematics 128.1-2 (2001), pp. 335–362. doi: 10.1016/s0377-0427(00)00518-5

    2001 doi

  40. [48]

    The method of alternating projection s and the method of subspace corrections in Hilbert space

    J. Xu and L. Zikatanov. “The method of alternating projection s and the method of subspace corrections in Hilbert space”. In: Journal of the American Mathematical Society 15.03 (2002), pp. 573–598. doi: 10.1090/s0894-0347-02-00398-3

    2002 doi

  41. [130]

    doi: 10.1137/1.9781611971057.ch4

    doi

  42. [680]

    doi: 10.1002/nla.1979

    1979 doi

  43. [2802]

    doi: 10.1137/080730044

    doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.