On $hp$-adaptive Structure-Preservion for the Cahn--Hilliard--Navier--Stokes Equations with Degenerate Mobility

Jimmy Kornelije Gunnarsson; Robert Kl\"ofkorn

REVIEW 4 major objections 6 minor 10 references

On $hp$-adaptive Structure-Preservion for the Cahn--Hilliard--Navier--Stokes Equations with Degenerate Mobility

T0 review · 4 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper claims that adding a parameterized, edge-wise mobility flux to two discontinuous Galerkin discretizations makes the Cahn-Hilliard-Navier-Stokes system coercive—and hence stable—under a computable condition, while preserving mass,

desk verdict New mobility-flux family is real but thin; coercivity proof hinges on an unverified contrast bound and structure-preservation claims outrun the proofs. read the letter →

arxiv 2602.22861 v3 pith:E4AIQYX4 submitted 2026-02-26 math.NA cs.NAmath-phmath.MP

Jimmy Kornelije Gunnarsson , Robert Klöfkorn This is my paper

classification math.NAcs.NAmath-phmath.MP MSC 65M6065N3065M5076D0535Q35

keywords Cahn-HilliardNavier-StokesdegeneratemobilitydiscontinuousGalerkincoercivitystructurepreservationhp-adaptivityphasefield

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 aims to establish that two new discontinuous Galerkin discretizations for the Cahn-Hilliard-Navier-Stokes equations with degenerate mobility are coercive and structure-preserving. The central claim is that a parameterized, edge-wise mobility flux—using either a harmonic average or an intersection maximum of the mobility—yields a provable coercivity bound for the discrete trilinear form, provided a penalty parameter exceeds a solution-dependent contrast ratio. Building on that coercivity, the authors argue for optimal convergence, discrete mass conservation, energy dissipation, and preservation of the phase-field bound, and they demonstrate the schemes on hp-adaptive meshes. A sympathetic reader would care because degenerate-mobility phase-field systems are difficult to keep bounded and stable in finite-element discretizations, and the new parameter α gives a concrete stability-control knob.

What carries the argument

The load-bearing object is the trilinear form b(M(ψ_h), v, φ) = ∫ M∇v·∇φ dx + penalty − flux terms, where the mobility flux F is either the harmonic average ⟨M⟩ (SWIP) or the arithmetic average {M∇v} (SIPG). The new ingredient is the parametrized edge mobility Λ_e(M) = β tilde M_e^{2α}, with tilde M_e the harmonic average or intersection maximum, together with the trace inequality of Lemma 2 for mixed polynomial order, which converts edge jumps into volume norms with the constant max p(p+d−1). This machinery turns the choice of α into a direct tradeoff: larger α gives a more diffusive edge flux that is easier to handle at intersections, at the cost of a potentially larger global contrast λ*.

What would settle it

Run one of the reported simulations (e.g., Ex. 2 or 3) and compute λ* = max_K Σ_{e∈∂K} ||tilde M_e||^{2−2α}_{L∞(e)} / min_{x∈K} M(ψ_h(x)) at representative times; if λ* exceeds the chosen β (3 or 5) on any element for the given δ=10^-20, the coercivity condition in Theorem 2 is not satisfied by that run.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is Theorem 2: for a fixed phase field ψ_h whose mobility is bounded below by δ, the generalized trilinear form b(M(ψ_h), v_h, v_h) is coercive with respect to a DG seminorm whenever η_e ≥ max_{p∈{p_{K_e^-}, p_{K_e^+}}} p(p+d−1) and Λ_e(M(ψ_h)) ≥ tilde M_e^{2α}(ψ_h) λ*. Here λ* is the maximum over elements of a ratio of edge mobility averages to the minimum of the mobility inside the element, and α ∈ [0,1/2] interpolates between the existing α=0 fluxes and the new, more diffusive α=1/2 fluxes. From this coercivity the paper derives optimal convergence, structure preservation, and a discrete maximum principle, and it validates these properties numericall

Load-bearing premise

The load-bearing premise is the a-priori bound β ≥ λ*, where λ* is a solution-dependent contrast ratio that can blow up as the mobility M(ψ) approaches zero; the paper assumes this bound (with β=3 or 5) rather than proving it, and the simulations use a mobility floor of 10^-20, so λ* may be enormous and Theorem 2 may not apply to the actual runs.

Editorial extensions

If this is right

If the coercivity condition holds, the SWIPD-L and SIPGD-L schemes inherit optimal convergence rates in L2 and H1, confirmed numerically in 2D and 3D.
Mass conservation, energy dissipation, and discrete boundedness of the phase field are preserved, which the numerical experiments for droplet merging and rotating bubbles corroborate.
The α=1/2 'D' flux is more diffusive and improves treatment over element intersections, at the price of possibly larger λ*.
The hp-adaptive formulation achieves quality comparable to h-adaptive runs with significantly fewer degrees of freedom.
The two flux variants produce identical convergence rates, since the mobility flux affects stability and coercivity, not consistency.

Reading between the lines

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

The theorem's condition β ≥ λ* is assumed rather than verified; a direct post-processing of the reported runs could check whether β=3 or 5 actually exceeds the computed λ*, and if not, the observed stability would stand on different, unproven grounds.
If the parameter α can be tuned adaptively per element or per interface, the coercivity condition suggests a practical way to reduce over-penalization while preserving stability on adaptive meshes.
The harmonic-average flux naturally feeds into a limit argument for a discrete maximum principle; the paper lists this as future work, and Lemma 2 supplies the missing mixed-order trace ingredient.

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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

Quick take: this is an incremental DG paper that introduces an α-parameterized mobility flux for the Cahn-Hilliard-Navier-Stokes penalty term and proves a conditional coercivity bound. The novelty is genuine but narrow—the authors themselves call the schemes an algebraic reformulation of their earlier SWIP-L/SIPG-L with a different penalty flux. What's solid: the convergence tables are clean, the rates match expectation in 2D and 3D, and the hp-adaptive implementation is real. The main theorem is a natural generalization of [8]'s coercivity argument to α∈[0,1/2].

The soft spots are real and need attention. First, Theorem 2's sufficient condition involves λ*, a solution-dependent contrast that can blow up as mobility degenerates. With δ=10^-20 in the regularization, λ* can be enormous, and the paper simply sets β=3 or 5 and assumes β≥λ* (Remark 1). No computation or bound of λ* is provided for Examples 2-3, so the proof does not demonstrably cover the actual runs. That's a gap between the headline claim and the numerics, not a sign the method is wrong.

Second, the structure-preservation language overreaches. The abstract says the methods preserve mass, energy, and the maximum principle, but the rigorous structure-preserving proofs are deferred to future work (Section 4); the energy/mass arguments are cited to [8], and the DMP is only empirical. The convergence tables show identical errors for the new schemes and the old ones, so the new penalty flux buys no accuracy advantage—only a claimed stability control that remains unverified.

Third, the 'significant computational savings' from hp-adaptivity are never quantified with degrees of freedom or runtime counts; we just get a qualitative statement.

The reliance on the authors' own [8] for the limiter, the adaptive indicator, and the energy/mass framework is heavy, but not a red flag by itself; it does mean the incremental distance from [8] is small.

Bottom line: a useful contribution for DG/CHNS practitioners, but it needs a major revision before it deserves to stand as claimed. The authors should either verify λ* for the reported tests, or state honestly that the coercivity theorem is conditional and the scheme is an empirical structure-preserving method. They should also quantify the adaptivity savings and fix the abstract. This deserves a serious referee—the math is coherent and the numerical work is careful—but I would not accept it in current form. I'd read it, and possibly use the α-family if the constants get resolved, but I wouldn't cite it as is.

Referee Report

4 major / 6 minor

Summary. The paper proposes two discontinuous Galerkin variants, SWIPD-L and SIPGD-L, for the Cahn-Hilliard--Navier--Stokes system with degenerate mobility. The methods extend the authors' earlier SIPG-L/SWIP-L schemes by introducing parametrized mobility fluxes depending on a parameter alpha; for alpha=1/2 the flux is more diffusive, for alpha=0 it reduces to the earlier schemes. The main theoretical claim is Theorem 2, a conditional coercivity result for the trilinear form b, requiring a solution-dependent maximum-contrast bound lambda*. The numerical part reports optimal convergence rates in 2D and 3D, mass/energy/boundedness preservation, and hp-adaptivity savings. The central issue is that the coercivity condition is never verified for the actual simulations, and the paper itself defers rigorous discrete structure-preserving proofs to future work.

Significance. If the coercivity theorem were fully established, the alpha-parametrized flux family would be a useful, tunable extension of [8] with a clear stability--consistency trade-off. The numerical experiments are well organized, the convergence tables are clean and consistent with expected rates, and the hp-adaptive tests show a genuine reduction in degrees of freedom. However, the theoretical anchor is conditional on an unverified solution-dependent constant, and several structure-preservation properties are only inherited from [8] or observed numerically. The paper is therefore best viewed as an empirical scheme with a partial theory, not as a proof of a new structure-preserving method in its current form.

major comments (4)

[Theorem 2, Eqs. (14)-(15), Remark 1, Section 3] The coercivity theorem requires Lambda_e >= \tilde M_e^{2alpha} lambda*, with lambda* a solution-dependent maximum contrast defined in Eq. (15). Remark 1 replaces this by Lambda_e = beta \tilde M_e^{2alpha} with fixed beta=3 or 5 and simply assumes beta >= lambda*. The manuscript never computes lambda* or gives a bound for Exs. 2--3. With the regularization delta=1e-20, min M can be as small as delta near the bounds, making lambda* potentially O(1e20). For example, the simulations use beta=3 while the theorem needs beta >= lambda*. Thus the implemented scheme is not demonstrated to satisfy the theorem's sufficient condition, and the claim 'provably coercive' is not supported for the actual runs. Please either verify the condition (e.g., by reporting computed lambda* values) or explicitly state that Theorem 2 is a conditional sufficient condition and soften the theoretical claims.
[Lemma 2, Eqs. (10)-(13)] Lemma 2 as stated is false for non-uniform meshes. Take |K^+|=1, |K^-|=4, |e|=1, C_{K^+}=C_{K^-}=C_e=1, and let phi^-=0. Then h_e=8/5, and Eq. (10) gives ||{nabla phi·n}||^2 <= 1/(2h_e) ||nabla phi^+||^2 = 5/16 ||nabla phi^+||^2, while the bound from (12) and Lemma 1 is 1/2 * |e|/|K^+| ||nabla phi^+||^2 = 1/2 ||nabla phi^+||^2. The proof's Eq. (13) underestimates the sum |e|/|K^+| C_{K^+} + |e|/|K^-| C_{K^-}; this sum is 2 C_e/h_e, not C_e/h_e. The error may be absorbed by the free penalty eta_e in Theorem 2, but as written the lemma does not support the proof. Please correct the constant and the proof.
[Theorem 2, Eq. (17)] The proof of Theorem 2 is only a sketch. Eq. (17) is asserted without derivation, and for both fluxes the factorization is not transparent. For SIPG, F = {M nabla upsilon} is an arithmetic average of a product, not a product of a maximum with an average; for SWIP, \tilde M_e = <M> and the exponent split <M>{nabla upsilon} = \tilde M^{1-alpha} ... is ambiguous. Since Theorem 2 is the main theoretical contribution, the complete proof with all constants and the precise application of Lemma 2 should be provided, not deferred to 'follows similarly to [8]'.
[Abstract, Section 2, Section 4] The abstract and introduction claim preservation of mass, energy dissipation, and the discrete maximum principle. In Section 2 the dissipation and conservation statements are only said to follow 'following similar arguments as in [8]'; no proof is given for the new fluxes in Eq. (7). Section 4 then states that 'rigorous structure-preserving proofs for the SWIPD-L scheme in the discrete setting' are future work. This is contradictory: if the discrete DMP and energy stability are not proven for the proposed scheme, they should be presented as numerical observations, not as theoretical properties. Please clearly delineate which statements are theorems, which are inherited, and which are empirical.

minor comments (6)

[Title] The arXiv title contains the typo 'Structure-Preservion'; the running title in the manuscript uses 'Structure-Preserving'.
[Eq. (14)] The set in the maximum should be over polynomial orders, e.g., p in {p_{K_e^+}, p_{K_e^-}}, not p in {K_e^+, K_e^-}.
[Eqs. (15)-(17)] The notation \tilde M_e and \tilde M is inconsistent. Please define the domain of each object and use a single symbol for the edge mobility flux in the statement of Theorem 2 and Eq. (17).
[Section 2.2, Fig. 1] The text refers to 'Fig. 2.2' when describing the limiter; this should be Fig. 1.
[Section 3 after Table 2] The text says beta=3 is used 'as in Rmk. 1', but Remark 1 gives beta=5 as the example from [8]; clarify which beta is used in each experiment.
[Table 1] The rows for SIPG-L, SWIP-L, SIPGD-L, and SWIPD-L are identical for the same p and N. This is consistent with the claim that the flux does not affect consistency, but the table could be condensed to one scheme per row with a note that the others agree to printed precision.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the coercivity theorem is a new derivation; self-citations to [8] and the unverified lambda* condition are correctness concerns, not circular reductions.

full rationale

The paper's central new claim, Theorem 2, is not a reformulation of its inputs. The coercivity condition (14)-(15) is derived through the trace inequalities in Lemmas 1-2 and the Cauchy-Schwarz bound (17), explicitly generalizing the proof of [8, Theorem 3.2] from alpha=0 to alpha in [0,1/2]. The SWIPD-L and SIPGD-L schemes are defined by their mobility flux choices, and the penalty condition is a standard sufficient condition for coercivity rather than a restatement of the definition. The structure-preserving claims (mass conservation, energy dissipation, boundedness) are largely deferred to the same authors' preprint [8] (e.g., 'Following similar arguments as in [8] and references therein' and 'The outline and algorithm of the scheme are very similar to those detailed in [8]'); this is self-citation that is load-bearing for those ancillary theoretical statements, but the numerical experiments independently verify these properties for the reported cases. The substitution in Remark 1 of a fixed beta >= lambda* for the solution-dependent lambda* of (15), with no computation of lambda* under delta=1e-20, means the theorem's hypothesis is not verified for the actual runs; this is an unverified assumption and a correctness gap, but it is not a circular reduction. No fitted quantity is relabeled as a prediction, and no self-citation is used to forbid alternatives. Therefore no significant circularity is found, and the score reflects the residual self-citation concern only.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central method rests on several under-supported ingredients: a solution-dependent contrast λ* that is never bounded, a fixed β assumed to dominate it, a δ-regularization of the degenerate mobility, and inherited structure-preservation claims from the same authors' [8]. The α-parameterized flux family and conditional coercivity estimate are new, but the ledger shows the reader ultimately pays for the stability condition via an a-priori guess and numerical observation.

free parameters (5)

β (mobility flux contrast estimate) = β=3 in this paper; β=5 in [8]
Used in coercivity condition Λ_e = β \tilde M_e^{2α}; assumed β≥λ* though λ* is never computed; if the assumption fails, the coercivity theorem does not apply.
δ (mobility regularization floor) = 10^-20
Replaces degenerate mobility M by max{M,δ} so Theorem 2's 'M≥δ' hypothesis holds; means the method solves a non-degenerate perturbation of the target problem.
α (flux interpolation parameter) = 1/2 for SWIPD-L/SIPGD-L; 0 for SWIP-L/SIPG-L
Hand-chosen to interpolate between harmonic-average and arithmetic/max fluxes; controls the diffusivity of the penalty flux.
hp-adaptivity thresholds = H<0.0525 refine; H>0.15 coarsen; p-coarsening at H>0.075
Explicitly called 'somewhat arbitrary and can be tuned' in Section 2.2; the claimed hp-savings depend on these choices.
Nonlinear tolerance ε (Ex. 3) = 5e-16
Authors note mass drift scales with ε; chosen to avoid solver stagnation; affects reported mass conservation.

assumptions (6)

standard math Trace inequalities for DG interfaces (Lemma 1 from [9]; Lemma 2 for mixed polynomial orders)
Basis of the coercivity proof; Lemma 2 is derived in the paper but not formally machine-checked.
domain assumption Continuum weak maximum principle (Theorem 1 from [5])
Motivates the discrete boundedness target; assumes convex Ω, bounded initial phase, and finite initial energy; not transferable to the discrete scheme.
domain assumption Regularized mobility M_δ = max{M, δ} with δ=10^-20
Needed for the hypothesis M≥δ in Theorem 2; modifies the degenerate mobility, and the δ→0 limit is not analyzed.
ad hoc to paper β≥λ* with fixed β=3/5
Coercivity condition (14) requires a solution-dependent global contrast λ*; the paper substitutes an unproven a-priori constant.
ad hoc to paper Discrete energy dissipation and mass conservation inherited from [8]
No proof is provided in this paper; Section 4 defers rigorous structure-preserving analysis to future work.
ad hoc to paper Arbitrary refinement/coarsening thresholds in hp-adaptivity (0.0525, 0.15, 0.075)
Authors explicitly call them 'somewhat arbitrary and can be tuned'; they determine the reported computational savings.

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Cite this review

Pith. "Pith review of On $hp$-adaptive Structure-Preservion for the Cahn--Hilliard--Navier--Stokes Equations with Degenerate Mobility." pith.science (2026). https://pith.science/paper/E4AIQYX4

@misc{pith2026260222861,
  author       = {Pith},
  title        = {Pith review of: On $hp$-adaptive Structure-Preservion for the Cahn--Hilliard--Navier--Stokes Equations with Degenerate Mobility},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E4AIQYX4}},
  note         = {Machine review of arXiv:2602.22861}
}

abstract

We develop structure-preserving discontinuous Galerkin methods for the Cahn-Hilliard-Navier-Stokes equations with degenerate mobility. The proposed SWIPD-L and SIPGD-L methods incorporate parametrized mobility fluxes with edge-wise mobility treatments for enhanced coercivity-stability control. We prove coercivity for the generalized trilinear form and demonstrate optimal convergence rates while preserving mass conservation, energy dissipation, and the discrete maximum principle. Comparisons with existing SIPG-L and SWIP-L methods confirm similar stability. Validation on $hp$-adaptive meshes for both standalone Cahn-Hilliard and coupled systems shows significant computational savings without accuracy loss.

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Works this paper leans on

10 extracted references · 3 canonical work pages

[8]

DOI arXiv:2602.08639v2

Gunnarsson, J.K., Kl ¨ofkorn, R.: Comparison of Structure Preserving Schemes for the Cahn- Hilliard-Navier-Stokes Equations with Degenerate Mobility and Adaptive Mesh Refinement (2026). DOI arXiv:2602.08639v2. URLhttps://arxiv.org/abs/2602.08639

arXiv 2026
[1]

IMA Journal of Numerical Analysis31(1), 254–280 (2009)

Ainsworth, M., Rankin, R.: Constant free error bounds for nonuniform order discontinuous Galerkin finite-element approximation on locally refined meshes with hanging nodes. IMA Journal of Numerical Analysis31(1), 254–280 (2009). DOI 10.1093/imanum/drp025

work page doi:10.1093/imanum/drp025 2009
[2]

CAMWA81, 75–112 (2021)

Bastian, P.,et al.: The Dune framework: Basic concepts and recent developments. CAMWA81, 75–112 (2021). DOI 10.1016/j.camwa.2020.06.007

doi:10.1016/j.camwa.2020.06.007 2021
[3]

Dedner, A., Kl ¨ofkorn, R.: Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Method. Commun. Appl. Math. Comput. (2021). DOI 10.1007/s42967-021-00134-5

work page doi:10.1007/s42967-021-00134-5 2021
[4]

DOI 10.5281/zenodo.3706994

Dedner, A., Kl ¨ofkorn, R., Nolte, M.: Python bindings for the dune-fem module (2020). DOI 10.5281/zenodo.3706994

doi:10.5281/zenodo.3706994 2020
[5]

Siam Journal on Mathematical Analysis27(2), 404–423 (1996)

Elliott, C.M., Garcke, H.: On the Cahn–Hilliard equation with degenerate mobility. Siam Journal on Mathematical Analysis27(2), 404–423 (1996). DOI 10.1137/S0036141094267662

doi:10.1137/s0036141094267662 1996
[6]

Journal of Computational and Applied Mathematics206(2), 843–872 (2007)

Epshteyn, Y., Rivi`ere, B.: Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics206(2), 843–872 (2007). DOI 10.1016/j.cam.2006.08.029

work page doi:10.1016/j.cam.2006.08.029 2007
[7]

IMA Journal of Numerical Analysis29(2) (2008)

Ern, A., Stephansen, A.F., Zunino, P.: A discontinuous galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA Journal of Numerical Analysis29(2) (2008). DOI 10.1093/imanum/drm050

doi:10.1093/imanum/drm050 2008

Show all 10 references

[9]

SIAM (2008)

Rivi `ere, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. SIAM (2008). DOI 10.1137/1.9780898717440

2008 doi
[10]

Journal of Computational Physics229(9), 3091–3120 (2010)

Zhang, X., Shu, C.W.: On maximum-principle-satisfying high order schemes for scalar conserva- tion laws. Journal of Computational Physics229(9), 3091–3120 (2010). DOI 10.1016/j.jcp.2009. 12.030

2010 doi

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [8]

DOI arXiv:2602.08639v2

Gunnarsson, J.K., Kl ¨ofkorn, R.: Comparison of Structure Preserving Schemes for the Cahn- Hilliard-Navier-Stokes Equations with Degenerate Mobility and Adaptive Mesh Refinement (2026). DOI arXiv:2602.08639v2. URLhttps://arxiv.org/abs/2602.08639

arXiv 2026

[2] [1]

IMA Journal of Numerical Analysis31(1), 254–280 (2009)

Ainsworth, M., Rankin, R.: Constant free error bounds for nonuniform order discontinuous Galerkin finite-element approximation on locally refined meshes with hanging nodes. IMA Journal of Numerical Analysis31(1), 254–280 (2009). DOI 10.1093/imanum/drp025

work page doi:10.1093/imanum/drp025 2009

[3] [2]

CAMWA81, 75–112 (2021)

Bastian, P.,et al.: The Dune framework: Basic concepts and recent developments. CAMWA81, 75–112 (2021). DOI 10.1016/j.camwa.2020.06.007

doi:10.1016/j.camwa.2020.06.007 2021

[4] [3]

Dedner, A., Kl ¨ofkorn, R.: Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Method. Commun. Appl. Math. Comput. (2021). DOI 10.1007/s42967-021-00134-5

work page doi:10.1007/s42967-021-00134-5 2021

[5] [4]

DOI 10.5281/zenodo.3706994

Dedner, A., Kl ¨ofkorn, R., Nolte, M.: Python bindings for the dune-fem module (2020). DOI 10.5281/zenodo.3706994

doi:10.5281/zenodo.3706994 2020

[6] [5]

Siam Journal on Mathematical Analysis27(2), 404–423 (1996)

Elliott, C.M., Garcke, H.: On the Cahn–Hilliard equation with degenerate mobility. Siam Journal on Mathematical Analysis27(2), 404–423 (1996). DOI 10.1137/S0036141094267662

doi:10.1137/s0036141094267662 1996

[7] [6]

Journal of Computational and Applied Mathematics206(2), 843–872 (2007)

Epshteyn, Y., Rivi`ere, B.: Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics206(2), 843–872 (2007). DOI 10.1016/j.cam.2006.08.029

work page doi:10.1016/j.cam.2006.08.029 2007

[8] [7]

IMA Journal of Numerical Analysis29(2) (2008)

Ern, A., Stephansen, A.F., Zunino, P.: A discontinuous galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA Journal of Numerical Analysis29(2) (2008). DOI 10.1093/imanum/drm050

doi:10.1093/imanum/drm050 2008