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REVIEW 4 major objections 3 minor 34 references

On Finite Element Methods for Heterogeneous Elliptic Problems

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

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Pith's one-line read The paper claims that exactly imposing potential continuity, normal-flux continuity, and the tangential resistivity-velocity condition on a material interface restores optimal finite element convergence rates for heterogeneous Darcy flow.

desk verdict Useful reprint of a 2008 stabilized-mixed interface method, but the stability proof leans on an unproved uniform lower bound for the interface transformation and the numerics only test γ=1.0. read the letter →

arxiv 2506.08251 v1 pith:GELPLV7O submitted 2025-06-09 math.NA cs.NA

Abimael F. D. Loula , Maicon R. Correa , João N. C. Guerreiro , Elson M. Toledo This is my paper
classification math.NAcs.NA MSC 65N1265N2265N3035A35
keywords stabilizedmixedfiniteelementmethodsdiscontinuousGalerkininterfaceconditionsleastsquaresheterogeneousmediaDarcyflowanisotropicconductivityconvergencerates
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

The paper sets out to show that stabilized mixed finite element methods for Darcy flow can be made to work in heterogeneous media with a material interface, where the velocity field is discontinuous. It first documents that standard continuous ($C^0$) Lagrangian stabilized methods — MGLS, HVM, and CGLS — lose accuracy on such problems because they force the tangential velocity to be continuous across the interface, while the physics makes it jump. The proposed fix is a constrained formulation that exactly imposes the correct interface conditions (potential continuity, normal-flux continuity, and continuity of the tangential component of resistivity times velocity) through an element-level linear transformation. The paper argues that this recovers stability, accuracy, and the optimal convergence rates of classical Lagrangian interpolation, and its numerical studies on an anisotropic two-material test problem confirm those rates. A reader should care because reservoirs and composite materials are exactly the situation where continuous mixed elements currently fail.

What carries the argument

The load-bearing mechanism is the linear transformation $\pi: U_h^l\times P_h^k \to U_{h0}^l\times P_{h0}^k$ that enforces the interface constraints by mapping discontinuous subdomain degrees of freedom into a continuous reference pair $(\bar u_h,\bar p_h)$. On interface nodes the $\Omega_1$ velocity unknowns are replaced by $[Q_1]^{-1}[Q_2]$ times the $\Omega_2$ unknowns, where $[Q_i]$ is assembled from the resistivity tensor $\Lambda_i$ and the normal/tangent vectors; this is exactly what makes $\Lambda_1 u_1\cdot\tau=\Lambda_2 u_2\cdot\tau$ hold while allowing $u_1\cdot\tau\neq u_2\cdot\tau$. Element matrices of interface elements are transformed as $\bar K^e=[T]^T K^e [T]$, preserving symmetry and the usual $C^0$ connectivity. The CGLS stabilization supplies the underlying inf-sup stability, which the paper transfers to the constrained problem by asserting that $\pi$ is continuous and bounded below.

What would settle it

Compute the discrete inf-sup constant (or minimum singular value of the transformed CGLS$\pi$ matrix) on the two-material test problem while refining the mesh and varying the conductivity ratio $\gamma$ from $10^{-4}$ to $10^4$. If the constant decays with $h$ or with the anisotropy contrast, the asserted $h$- and contrast-independent bound $\bar\beta>0$ in inequality (53) fails. A cheaper numerical check is to run the same Q1 convergence study with $\gamma=10^3$: if the reported $O(h^2)$ velocity rate degrades toward $O(h^{0.5})$, the optimal-rate claim is false.

Watch

Extended reading notes

Core claim

The paper's central claim is that the failure of $C^0(\Omega)$ Lagrangian stabilized mixed methods on heterogeneous elliptic problems is not inherent to the stabilized approach but comes from imposing the wrong interface continuity. For Darcy flow with conductivities $K_1\neq K_2$, the exact solution satisfies $u_1\cdot n=u_2\cdot n$ and $\Lambda_1 u_1\cdot\tau=\Lambda_2 u_2\cdot\tau$ with $\Lambda_i=K_i^{-1}$, while $u_1\cdot\tau\neq u_2\cdot\tau$; a continuous velocity interpolation enforces $u_h\cdot\tau$ continuous and therefore approximates an intermediate value, degrading convergence to about $O(h^{0.5})$ or worse. The paper constructs a constrained stabilized formulation (CGLS$\pi$) on spaces that are discontinuous across the interface, then maps them into a continuous reference space by a linear transformation $\pi$ that strongly imposes the three interface conditions at element level. After solving, the true discontinuous pair is recovered elementwise. The paper reports that on the anisotropic heterogeneous test problem the constrained MGLS, HVM, and CGLS methods recover the same optimal rates predicted for homogeneous smooth solutions — e.g. $O(h^{k+1})$ for potential and velocity with equal-order Q1/Q2 CGLS — while preserving normal flux continuity.

Load-bearing premise

The load-bearing premise is that the linear transformation gluing the two subdomain solutions at the interface has a stability constant that stays positive independently of mesh refinement and of how strong the conductivity jump is, a fact the paper asserts in Section 5.4 without proof.

Editorial extensions

If this is right

  • The same stabilized mixed formulations used for homogeneous Darcy flow can be applied to layered heterogeneous media by adding a purely algebraic transformation on interface elements, with no mesh-dependent parameters.
  • Optimal rates for potential, velocity, and divergence are recovered for piecewise-regular solutions with smooth interfaces, so equal-order Q1 and Q2 elements become reliable for heterogeneous reservoirs.
  • Normal flux continuity is preserved strongly on the interface, satisfying mass conservation, while the tangential component is allowed to jump exactly as Darcy's law requires.
  • Because the interface constraints are imposed at element level and the global problem has the same connectivity as a $C^0$ formulation, the approach avoids the degrees-of-freedom explosion of fully discontinuous Galerkin methods.
  • The numerical results further indicate that the constrained formulation removes the spurious oscillations and accuracy loss seen in the unconstrained continuous methods on the anisotropic test problem.

Reading between the lines

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

  • Beyond the paper: the same element-level transformation should extend to Stokes–Darcy coupling and other interface-constrained problems where a linear trace relation between subdomains is known.
  • Beyond the paper: the stability of $\pi$ needs a mesh- and contrast-independent lower bound; computing the discrete inf-sup constant on the two-material test over a range of conductivity ratios would settle whether the unproved bound in Section 5.4 is cosmetic or real.
  • Beyond the paper: the exactness of the tangential constraint depends on the interface being represented by element edges; curved interfaces approximated by piecewise-linear edges would introduce a geometric error that may reduce the observed rates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper reviews C0 Lagrangian stabilized mixed methods (MGLS, HVM, CGLS) for Darcy flow, demonstrates numerically that they lose their optimal convergence rates on a heterogeneous two-material problem with a discontinuous tangential velocity, and proposes a continuous/discontinuous formulation in which the interface conditions on potential, normal flux, and resistivity-weighted tangential velocity are imposed strongly through a nodal transformation π. The central claim, stated in Section 6, is that this constrained formulation recovers stability, accuracy, and the optimal convergence rates of the homogeneous problem. The stability argument in Section 5.4 transfers the homogeneous CGLS stability result to the constrained formulation via an assumed uniform lower bound on π.

Significance. The variational derivations in Sections 3 and 4 are standard and clearly presented, and the numerical study uses an external exact solution from Crumpton et al. (1995) with systematic mesh refinement, which is a genuine strength. The proposed element-level imposition of interface constraints is attractive because it preserves the sparsity structure of a C0 formulation. However, the theoretical core of the paper is an unproved lower bound on the transformation π in Section 5.4, and the numerical evidence covers only γ=1.0. If the stability assumption can be proved or replaced by a precise hypothesis, the method would be of practical interest; in its current form the advertised optimal-rate claim is not supported for strongly contrasting media.

major comments (4)
  1. [Section 5.4, Eq. (53)] The lower bound in Eq. (53) is the only link between the homogeneous CGLS stability result (51) and the constrained problem, but no proof or reference is given for it. For the test problem of Section 3.3, with K1=I and K2=γ[[2,1],[1,2]], the nodal transformation at the interface x=0 is [Q1]^{-1}[Q2]=[[1,0],[-1/(3γ),2/(3γ)]], whose smallest singular value is O(1/γ) as γ→∞. Consequently a constant β̄>0 independent of the conductivity contrast cannot exist, and even a contrast-dependent β̄ would require an explicit estimate. Since the numerical experiments in Section 5.5 use only γ=1.0, the stability of Problem CGLSπ for strongly heterogeneous media is left open.
  2. [Section 5.4, Eq. (51)] Equation (51) is attributed to Correa and Loula (2008) for the homogeneous C0 CGLS method, but the displayed inequality involves Bπ, with π applied to both trial and test, and the supremum is taken over continuous test functions. The homogeneous inf-sup property provides a test function in the discontinuous space U_h, not necessarily in the range of π. To conclude (51) one needs, in addition to the lower bound on ||πu||, that the range of π is stable for the homogeneous form and that test functions in that range can be pulled back with controlled norm. Neither condition is stated or proved.
  3. [Section 5.5 and Section 6] The convergence study in Section 5.5 is performed only for γ=1.0. The interface condition Λ1u1·τ=Λ2u2·τ and the transformation [Q1]^{-1}[Q2] depend explicitly on γ, and the degeneracy identified in Major Comment 1 occurs precisely as γ grows. The concluding claim that the method "recover[s] stability, accuracy and the optimal rates of convergence" is therefore not tested in the regime where the stability argument is most delicate. The authors should add experiments with large and small γ, or explicitly restrict the claim to moderate contrast.
  4. [End of Section 5.4] The text states that "a complete numerical analysis of these formulations will be presented in a forthcoming paper." As a result, the present manuscript contains no a priori error estimate for the constrained formulation, and the rates reported in Section 5.5 are numerical observations rather than proven statements. The concluding remarks should be softened accordingly, or the missing analysis should be included.
minor comments (3)
  1. [Sections 5.2 and 5.3] The direction of π is stated inconsistently: Section 5.2 says π maps U_h to U_h0, but Eq. (47) and the construction in Section 5.3 map the continuous reference solution to the discontinuous one, i.e., π: U_h0 → U_h. Please make the direction consistent, as it affects the reading of Eqs. (51) and (53).
  2. [Section 5.4] There is a typo: "bounded bellow" should be "bounded below"; the dedication also contains "Bevilaccqua" for "Bevilacqua".
  3. [Figures 7–9] The convergence plots label slopes such as 1.5, 2, and 3, but no table of measured errors is provided. Stating the computed slopes numerically would make the claimed O(h^2) and O(h^3) rates easier to verify.

Circularity Check

0 steps flagged · score 2.0 of 10

No structural circularity; the only same-author dependency is a minor, non-load-bearing stability citation, and the unproved bound in Eq. (53) is a proof-gap concern rather than a self-referential reduction.

full rationale

Walking the derivation chain: the interface transformation π in Section 5.3 is constructed so that the discrete space exactly satisfies the physical interface conditions (41)-(43); these are derived from the model, not fitted to the benchmark, and the convergence study in Section 5.5 is checked against the external exact solution of Crumpton et al. (1995). No fitted parameter is renamed as a prediction and no equation is equivalent to its own input. The only same-author dependency is Eq. (51), where stability "is proved in Correa and Loula (2008)"; that cited result applies to the homogeneous C0 CGLS block and has independent content. The extension to the constrained formulation additionally requires Eq. (53)'s unproved lower bound β̄ > 0, and the paper itself flags the incomplete analysis in Section 5.4 with "A complete numerical analysis of these formulations will be presented in a forthcoming paper." A missing h- and contrast-independent β̄ estimate is a correctness risk, not circularity, because the central claim is not reduced to the cited theorem alone and the numerical claim is externally benchmarked. Therefore no circular step is exhibited; the score reflects only the minor, non-load-bearing self-citation in the stability discussion.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard finite element approximation theory for smooth solutions on each subdomain, on the assumption of a smooth material interface with piecewise regular solution, and on the stability of the homogeneous CGLS method imported from a same-author reference. The transfer of stability to the constrained method additionally assumes, without proof, that the interface transformation π is bounded below uniformly in mesh size and material contrast. No new physical entities are proposed. The MGLS stabilization parameters are chosen, not fitted.

free parameters (1)
  • MGLS stabilization parameters δ1, δ2 = δ1 = δ2 = 1/2
    Chosen for the numerical convergence study of MGLS in Section 3.3; the MGLS method only requires δ1 > 0 and δ2 > 0, and the reported convergence rates are not sensitive to this choice. Not fitted to data.
assumptions (4)
  • standard math Lax-Milgram lemma and standard H^1 conforming finite element error estimates (Ciarlet, 1978)
    Used in Section 3.1 to establish existence, uniqueness, and the O(h^{k+1}) potential and O(h^k) velocity estimates that motivate the rates discussed in Section 3.3.
  • standard math Stability of the homogeneous CGLS method in the sense of Babuska, proved in Correa and Loula (2008)
    The stability inequality (51) is imported from Correa and Loula (2008) and is the foundation for the constrained stability argument in Section 5.4.
  • domain assumption Smooth interface Γ and piecewise constant, positive definite conductivity tensors K_i with a piecewise regular exact solution
    The model problem in Section 2 and the convergence study in Section 5.5 assume the interface is smooth and the solution is H^{k+1} in each subdomain, which is exactly the setting where homogeneous rates are expected.
  • ad hoc to paper The linear transformation π is continuous and bounded below uniformly in h, so β̄ > 0 exists in inequality (53)
    Invoked in Section 5.4 to transfer stability from the homogeneous CGLS method to the constrained problem; no proof or independent reference is supplied for this bound.

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Cite this review

Pith. "Pith review of On Finite Element Methods for Heterogeneous Elliptic Problems." pith.science (2026). https://pith.science/paper/GELPLV7O

@misc{pith2026250608251,
  author       = {Pith},
  title        = {Pith review of: On Finite Element Methods for Heterogeneous Elliptic Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GELPLV7O}},
  note         = {Machine review of arXiv:2506.08251}
}
read the original abstract

Dealing with variational formulations of second order elliptic problems with discontinuous coefficients, we recall a single field minimization problem of an extended functional presented by Bevilacqua et al (1974), which we associate with the basic idea supporting discontinuous Galerkin finite element methods. We review residual based stabilized mixed methods applied to Darcy flow in homogeneous porous media and extend them to heterogeneous media with an interface of discontinuity. For smooth interfaces, the proposed formulations preserve the continuity of the flux and exactly imposes the constraint between the tangent components of Darcy velocity on the interface. Convergence studies for a heterogeneous and anisotropic porous medium confirm the same rates of convergence predicted for homogeneous problem with smooth solutions.

Figures

Figures reproduced from arXiv: 2506.08251 by the authors.

Figure 1
Figure 1. Heterogeneous media with an interface of discontinuity. For simplicity we will consider the porous media homogeneous and anisotropic on each subdomain. The mass balance on each subdomain is established as div u1 “ f1 in Ω1 (1) div u2 “ f2 in Ω2 (2) where fi “ f|Ωi are sources of fluid and ui “ u|Ωi , i “ 1, 2 are average velocities of the fluid in each domain Ωi , given by Darcy’s law u1 “ ´ K1 ∇p1 in Ω1 (3) [PITH_… view at source ↗
Figure 2
Figure 2. Component uy; (a) Exact and (b) approximated by HVM with 8 ˆ 8 bilinear elements. Oph 0.5 q are obtained for HVM and MGLS methods for both Q1 and Q2 elements, while no convergence is observed for the CGLS method as a consequence of the lack of global regularity of the exact solution combined with the use of continuous interpolations to approximate a discontinuous field. -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.8 1 1.2 1.4 1.… view at source ↗
Figure 3
Figure 3. Convergence study for an anisotropic heterogeneous problem and continuous approximations; Velocity with (a) bilinear elements and (b) biquadratic elements [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Convergence study for an anisotropic heterogeneous problem and continuous approximations; Divergence of velocity with (a) bilinear elements and (b) biquadratic elements. Figures 4a and 4b present convergence results for the divergence of the velocity ap￾proximations. T…
Figure 5
Figure 5. Figure 5: Convergence study for an anisotropic heterogeneous problem and continuous approximations; Potential with (a) bilinear elements and (b) biquadratic elements. 4. Broken Space Formulations In this Section we comment on special single field formulations of our model proble…
Figure 6
Figure 6. Figure 6: Component uy; (a) Exact and (b) approximated by CGLS with 8 ˆ 8 bilinear elements. Convergence results of velocity approximations for Q1 and Q2 elements with contin￾uous/discontinuous stabilizations are presented in Figures 7a and 7b, respectively. The CGLS and HVM met…
Figure 7
Figure 7. Figure 7: Convergence study for an anisotropic heterogeneous problem by capturing the discontinuity; Velocity with (a) bilinear elements and (b) biquadratic elements. of discontinuity. Stabilized mixed methods associated with C 0 Lagrangian interpolation present highly stable an…
Figure 8
Figure 8. Figure 8: Convergence study for an anisotropic heterogeneous problem by capturing the discontinuity; Divergence of velocity with (a) bilinear ele￾ments and (b) biquadratic elements. References Alvarez, G. B., Loula, A. F. D., Dutra do Carmo, E. D., Rochinha, F. A., 2006. A dis￾c…
Figure 9
Figure 9. Figure 9: Convergence study for an anisotropic heterogeneous problem by capturing the discontinuity; Potential with (a) bilinear elements and (b) biquadratic elements. Brezzi, F., Douglas Jr., J., Marini, L. D., 1985. Two families of mixed finite elements for second order ellipt…

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Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    B., Loula, A

    Alvarez, G. B., Loula, A. F. D., Dutra do Carmo, E. D., Rochinha, F. A., 2006. A dis- continuous finite element formulation for the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering 195, 4018–4035

    work page 2006

  2. [2]

    N., 1982

    Arnold, D. N., 1982. An interior penalty finite element method with discontinuous ele- ments. SIAM J. Numer. Anal. 19, 742–760

    work page 1982

  3. [3]

    N., Brezzi, F., Cockburn, B., Marini, D., 2002

    Arnold, D. N., Brezzi, F., Cockburn, B., Marini, D., 2002. A unified analysis of discon- tinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779. Babˇ uska, I., 1971. Error bounds for finite element method. Numerische Mathematik 16, 322–133

    work page 2002

  4. [4]

    P., Valentin, F., 2007

    Barrenechea, G., Franca, L. P., Valentin, F., 2007. A Petrov-Galerkin enriched method: A mass conservative finite element method for the Darcy equation. Computer Methods in Applied Mechanics and Engineering 196, 2449–2464

    work page 2007

  5. [5]

    A., Rojas, L

    Bevilacqua, L., Feij´ oo, R. A., Rojas, L. F., 1974. A variational principle for the Laplace operator with application in the torsion of composite rods. Int. J. Solids Structures 10, 1091–1102

    work page 1974

  6. [6]

    On the existence, uniqueness and approximation of saddle point problems arising from lagrange multipliers

    Brezzi, F., 1974. On the existence, uniqueness and approximation of saddle point problems arising from lagrange multipliers. RAIRO Analyse num´ erique/Numerical Analysis 8(R- 2), 129–151

    work page 1974

  7. [7]

    D., S¨ uli, E., 2006

    Brezzi, F., Cockburn, B., Marini, L. D., S¨ uli, E., 2006. Stabilization mechanisms in dis- continuous Galerkin finite element methods. Computer Methods in Applied Mechanics and Engineering 195, 3293–3310. ON FEM FOR HETEROGENEOUS ELLIPTIC PROBLEMS 21 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.8 1 1.2 1.4 1.6 1.8 log||error|| -log(h) Anisotropic Heterogeneous...

    work page 2006

  8. [8]

    D., 1985

    Brezzi, F., Douglas Jr., J., Marini, L. D., 1985. Two families of mixed finite elements for second order elliptic problems. Numerische Mathematik 47, 217–235

    work page 1985

Show all 34 references
  1. [9]

    Mixed and Hybrid Finite Element Methods

    Brezzi, F., Fortin, M., 1991. Mixed and Hybrid Finite Element Methods. Vol. 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York

    1991

  2. [10]

    Brezzi, F., Hughes, T. J. R., Marini, L. D., Masud, A., 2005. A mixed discontinuous Galerkin method for Darcy flow. SIAM J. Scientific Comput. 22-23, 119–145

    2005

  3. [11]

    G., 1978

    Ciarlet, P. G., 1978. The Finite Element Method for Elliptic Problems. Studies in Math- ematics and its Applications. North-Holland Publishing Company

    1978

  4. [12]

    Continuous groundwater velocity fields and path lines in linear, bilinear and trilinear finite elements

    Cordes, C., Kinzelbach, W., 1992. Continuous groundwater velocity fields and path lines in linear, bilinear and trilinear finite elements. Water Resources Research 28 (11), 2903– 2911

    1992

  5. [13]

    R., 2006

    Correa, M. R., 2006. Stabilized Finite Element Methods for Darcy and Coulped Stokes- Darcy Flows. D.Sc. Thesis, LNCC, Petr´ opolis, RJ, Brazil (in Portuguese)

    2006

  6. [14]

    R., Loula, A

    Correa, M. R., Loula, A. F. D., 2007. Stabilized velocity post-processings for Darcy flow in heterogenous porous media. Communications in Numerical Methods in Engineering 23, 461–489

    2007

  7. [15]

    R., Loula, A

    Correa, M. R., Loula, A. F. D., 2008. Unconditionally stable mixed finite element methods for Darcy flow. Computer Methods in Applied Mechanics and Engineering 197, 1525– 1540

    2008

  8. [16]

    I., Shaw, G

    Crumpton, P. I., Shaw, G. J., Ware, A. F., 1995. Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients. Journal of Computational Physics 116, 343–358. 22 A.F.D. LOULA, M.R. CORREA, J.N.C. GUERREIRO, AND ...

    1995

  9. [17]

    Coupling local discontinuous and continuous Galerkin methods for flow problems

    Dawson, C., 2005. Coupling local discontinuous and continuous Galerkin methods for flow problems. Water Resources Research 28, 729–744

    2005

  10. [18]

    Interior penalty proceedures for elliptic and parabolic Galerkin methods

    Douglas, J., Dupont, T., 1976. Interior penalty proceedures for elliptic and parabolic Galerkin methods. Lecture Notes in Physics 58, 207–216

    1976

  11. [19]

    Duarte, A. V. C., Rochinha, F. A., Dutra do Carmo, E. G., 2000. Discontinuous finite element formulations applied to cracked elastic domains. Computer Methods in Applied Mechanics and Engineering 185, 21–36

    2000

  12. [20]

    J., 1994

    Durlofsky, L. J., 1994. Accuracy of mixed and control volume finite element approxi- mations to Darcy velocity and related quantities. Water Resources Research 30 (4), 965–973. Dutra do Carmo, E. G., Duarte, A. V. C., 2002. New formulations and numerical analysis of discontinu...

    1994

  13. [21]

    P., Hughes, T

    Franca, L. P., Hughes, T. J. R., 1988. Two classes of mixed finite element methods. Computer Methods in Applied Mechanics and Engineering 69, 89–129

    1988

  14. [22]

    P., Hughes, T

    Franca, L. P., Hughes, T. J. R., Loula, A. F. D., Miranda, I., 1988. A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin method. Numerische Mathematik 53, 123–141

    1988

  15. [23]

    M., Fomin, S

    Gelfand, I. M., Fomin, S. V., 1963. Calculus of Variations. Prentice-Hall

    1963

  16. [24]

    H., S¨ uli, E., 2005

    Georgoulis, E. H., S¨ uli, E., 2005. Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method. IMA Journal of Numerical Anal- ysis 25, 205–220

    2005

  17. [25]

    Hughes, T. J. R., Masud, A., Wan, J., 2006. A stabilized mixed discontinuous Galerkin method for Darcy flow. Computer Methods in Applied Mechanics and Engineering 195, 3347–3381

    2006

  18. [26]

    Loula, A. F. D., Hughes, T. J. R., Franca, L. P., 1987. Mixed Petrov-Galerkin meth- ods for the Timoshenko beam problem. Computer Methods in Applied Mechanics and Engineering 63, 133–154

    1987

  19. [27]

    Loula, A. F. D., Rochinha, F. A., Murad, M. A., 1995. Higher-order gradient post- processings for second-order elliptic problems. Computer Methods in Applied Mechanics and Engineering 128, 361–381

    1995

  20. [28]

    Loula, A. F. D., Toledo, E. M., 1988. Dual and Primal Mixed Petrov-Galerkin Finite Element Methods in Heat Transfer Problems. LNCC - Technical Report 048/88

    1988

  21. [29]

    Masud, A., Hughes, T. J. R., 2002. A stabilized finite element method for Darcy flow. Computer Methods in Applied Mechanics and Engineering 191, 4341–4370. Mos´ e, R., Siegel, P., Ackerer, P., 1994. Application of the mixed hybrid finite element approximation in a groundwater ...

    2002

  22. [30]

    T., Babˇ uska, I., Baumann, C

    Oden, J. T., Babˇ uska, I., Baumann, C. E., 1998. A discontinuous hp finite element for diffusion problems. J. Comput. Phys. 146, 491–519

    1998

  23. [31]

    A., Thomas, J

    Raviart, P. A., Thomas, J. M., 1977. A mixed finite element method for second order elliptic problems. In: Math. Aspects of the F.E.M. No. 606 in Lectute Notes in Math- ematics. Springer-Verlag, pp. 292–315

    1977

  24. [32]

    Silva, R. C. C., Loula, A. F. D., 2002. Local residual error estimator and adaptive finite element analysis of Poisson’s problems. Computers & Structures 80, 2027–2034. ON FEM FOR HETEROGENEOUS ELLIPTIC PROBLEMS 23

    2002

  25. [33]

    M., 1990

    Toledo, E. M., 1990. New Mixed Finite Element Formulations With Post-Processing Schemes. D.Sc. Thesis, COPPE/UFRJ, Rio de Janeiro, RJ, Brazil (in Portuguese)

    1990

  26. [34]

    F., 1978

    Wheeler, M. F., 1978. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161

    1978

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