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Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems

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Pith's one-line read The paper proves that discontinuous Galerkin minimisers converge in L^p to minimisers of the quasiconvex-envelope (relaxed) energy, for quasiconvex and non-convex variational problems.

desk verdict New Gamma-convergence results for DG methods in quasiconvex/relaxed problems, but the stated penalty is not always well-defined and the proofs are not fully complete. read the letter →

arxiv 2501.11944 v1 pith:GFKO4MUI submitted 2025-01-21 math.NA cs.NAmath.AP

Georgios Grekas , Konstantinos Koumatos , Charalambos Makridakis , Andreas Vikelis This is my paper · ORCID
classification math.NAcs.NAmath.AP MSC 65N3049J4574B20
keywords discontinuousGalerkinquasiconvexityGamma-convergencerelaxationcalculusofvariationsnonlinearelasticityfiniteelementapproximationmicrostructure
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 proves that discontinuous Galerkin (DG) finite element methods converge for variational problems whose energy density $W$ is quasiconvex — the natural but weakest convexity notion in nonlinear elasticity — and, when $W$ is not convex, that discrete minimisers converge to minimisers of the relaxed problem $\int_\Omega W^{qc}$. This addresses a long-standing difficulty: DG approximations lack a true gradient structure, so the usual lower-semicontinuity argument for quasiconvex integrands does not apply directly. The proof restores the missing structure by reconstructing from each discontinuous $u_h$ a continuous piecewise polynomial $w_h$ whose error is controlled by the inter-element jumps. The result covers both the interior-penalty DG formulation of the authors' earlier work and the discrete-gradient formulation, and it is accompanied by numerical experiments on polyconvex and two-well (microstructure-forming) energies.

What carries the argument

The load-bearing object is a continuous reconstruction operator, generalised here from $p=2$ to any $p \geq 2$ (Lemma 2.2): for every discontinuous piecewise polynomial $u_h$ in the DG space $V_h^q$, there is a continuous piecewise polynomial $w_h$ with $\|\nabla^\alpha u_h - \nabla^\alpha w_h\|_{L^p}$ bounded by inter-element jump terms, uniformly in $h$. This reconstruction lets the proof pass from $u_h$ to $w_h$, where $\nabla w_h$ is a genuine gradient and $W^{qc}$ is lower semicontinuous under weak $L^p$ convergence. A second ingredient is the modified penalty $\mathrm{Pen}(u_h)$, built from $(1+|u_h|^p_{W^{1,p}})^{(p-1)/p}$ times the jump term, which provides coercivity for non-convex $W$ and makes the consistency terms vanish as $h \to 0$.

What would settle it

Choose a family of meshes where the reconstruction constant in Lemma 2.2 can be computed exactly (for example, a single row of elements with increasing aspect ratio) and check whether the constant stays bounded for $p>2$. If it unbounded while jump terms vanish, the liminf inequality has no uniform control. Alternatively, run the DG scheme on a two-well energy with boundary data that do not align with the laminate directions: if the computed $L^p$ limit is not the known relaxed minimiser, or the discrete energy does not approach $\min E^{qc}$, the convergence claim is false.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 3.1 and Theorem 3.2: for $W \in C^1$ satisfying $p$-growth and coercivity with $p>1$, any sequence of discrete minimisers $u_h$ of the DG energy $E_h$ converges in $L^p(\Omega)$, up to a subsequence, to a function $u$ that minimises the relaxed energy $E^{qc}(u)=\int_\Omega W^{qc}(\nabla u)$, and $E^{qc}(u)$ equals the infimum over the admissible set. When $W$ is quasiconvex, $W^{qc}=W$, so the discrete minimisers converge to genuine minimisers of the original problem. The theorem holds even though the discrete gradient is not a true gradient — its curl need not vanish — because the liminf inequality is obtained through the continuous reconstruction $w_h$ rather than through the discontinuous gradient $\nabla_h u_h$.

Load-bearing premise

Everything rests on the estimate that every discontinuous piecewise polynomial $u_h$ can be replaced by a continuous piecewise polynomial $w_h$ whose $L^p$ error, and the error of its gradient, is bounded by the size of the inter-element jumps uniformly in $h$; the paper gives the statement and a sketch but omits the full proof.

Editorial extensions

If this is right

  • For quasiconvex $W$, the discrete minimisers of the DG energy converge in $L^p(\Omega)$ to a minimiser of the continuous energy $\int_\Omega W(\nabla u)$, so the method can be trusted for elasticity-type problems without full convexity.
  • For non-quasiconvex $W$, the discrete minimisers converge to a minimiser of the relaxed energy $\int_\Omega W^{qc}(\nabla u)$, meaning that finitely many mesh oscillations are correctly averaged into the macroscopic relaxed state.
  • Both the interior-penalty DG formulation and the discrete-gradient DG formulation satisfy the same convergence theorem, unifying the DG methods previously known to converge only for convex energies.
  • The numerical two-well experiments show discrete minimising sequences forming ever-finer laminates whose $L^2$ limit is the relaxed minimiser, consistent with the $\Gamma$-convergence result.
  • The paper's pointwise-energy experiments suggest that quasiconvex envelopes $W^{qc}(F)$ can be approximated by solving a sequence of discrete minimisation problems.

Reading between the lines

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

  • A natural testable extension is to check whether the reconstruction estimates remain uniform on anisotropic or adaptively refined meshes; the paper's proof assumes a family where the constants stay bounded as $h\to 0$.
  • The same reconstruction-plus-liminf strategy could be applied to other relaxed objects such as polyconvex or rank-one convex envelopes, and to problems with Lavrentiev gaps, but the paper does not claim these extensions.
  • The paper's two-well numerics report roundoff degradation for large $p$ (the $p=8$ example), suggesting that a rescaled jump term is needed in practice; this is a practical adjustment the theory does not address.
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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

1 major / 4 minor

Summary. The manuscript develops a Gamma-convergence framework for discontinuous Galerkin discretizations of vector variational problems with quasiconvex and nonconvex energies. The two main theorems, Theorem 3.1 and Theorem 3.2, claim that sequences of discrete minimizers of two DG energy functionals, the method of [27] and the discrete-gradient method of [38], converge in L^p to minimizers of the relaxed energy E^{qc}(u)=∫ W^{qc}(∇u) dx, for W∈C^1 satisfying the growth, coercivity, and Lipschitz conditions (3.2)-(3.3). The proof strategy is: compactness from a coercivity estimate in Lemma 3.1; a liminf inequality obtained by reconstructing a continuous finite element function w_h from each discontinuous u_h (Lemma 2.2) and then applying quasiconvex lower semicontinuity; and a limsup (recovery sequence) argument via Dacorogna's relaxation theorem and interpolation estimates. The paper also reports numerical experiments for a polyconvex energy and for a two-well frame-indifferent energy, illustrating the behavior of the new penalty term and the emergence of microstructures.

Significance. If the stated results are correct, they are a significant advance: they remove the convexity requirement that had been essential in previous Gamma-convergence analyses of DG methods for variational problems, and they cover the two known convergent DG formulations. The continuous-reconstruction argument, used to control the non-vanishing curl of the discrete gradient, is a new and plausible mechanism for restoring lower semicontinuity in the quasiconvex setting. The numerical experiments are relevant and seem to support the qualitative claims. However, the paper is not yet acceptable in its current form: the penalty term defining the minimized functional is not well-defined for all W admitted by the stated assumptions, and the central reconstruction lemma is only sketched. Both are fixable, so I recommend a major revision rather than rejection.

major comments (1)
  1. [§2.0.2, Lemma 2.2] The functional minimized in Theorem 3.1 is not well-defined on V_h^q for every W satisfying (3.2). Condition (3.2) only gives W(ξ) ≥ c(-1+|ξ|^p) with c>0, so W(0) may be arbitrarily negative. For example, take W(ξ)=-M+(M+1)|ξ|^p with M>1; this W satisfies (3.2)-(3.3), but for |Ω|>1/M the base 1+Σ_K∫_K W(∇u_h)+Σ_e h_e^{1-p}||Ju_hK||^p_{L^p(e)} is negative when u_h≡0 and u_0=0, so Pen(u_h) is undefined in real arithmetic because the exponent (p-1)/p is a non-integer. Thus Theorem 3.1 states convergence of minimizers of a functional that is not even a function on the discrete space for admissible data. Remark 3.1's asserted equivalence with the penalty (3.8) is also false in this case: the claimed lower bound 1+|u_h|^p_{W^{1,p}(Ω,T_h)} ≲ 1+Σ_K∫_K W(∇u_h)+Σ_e h_e^{1-p}||Ju_hK||^p fails because the right-hand side can be negative. This is a load-bearing gap, since the proof genuinely uses the penalty in the form (3.8). A concrete fix is to add an explicit lower bound such as W≥0 (or W(0)≥0) to the hypotheses, or to replace the first factor in (3.6) by its positive part, and then to recheck the estimates of Lemmas 3.1 and 3.2 under that assumption.
minor comments (4)
  1. [§2, Eq. (3.3)] The Lipschitz condition (3.3) contains a typo: the last term in the factor multiplying |ξ1-ξ2| should be |ξ2|^{p-1}, not a second copy of |ξ1|^{p-1}.
  2. [§3.2, after Eq. (3.32)] The proof of the limsup inequality for Theorem 3.2 is delegated to 'adopting in similar fashion arguments from the previous section'. Since this is a second main theorem, the recovery sequence for E_{G,h} should be given in more detail, especially the control of the lifting term R_h(u_h) along the recovery sequence.
  3. [§3.3, Lemma 3.3] The smoothing bound (3.21), asserted for a sequence (u_δ)⊂C^∞(Ω̄) with |u_δ|_{W^{2,p}(Ω)} ≲ δ^{-1}|v|_{W^{1,p}(Ω)}, is stated without proof or citation; please provide a reference or a short justification.
  4. [§4, Eqs. (3.6) and (4.8)] The numerical section says that the computations use the penalty (3.6), but the implemented formula (4.8) rewrites the exponent and the second factor. Please clarify that the actual implemented penalty is (4.8), and mention whether this rewrite changes any of the theoretical requirements.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the quasiconvex Γ-convergence proof is a new argument; the penalty-equivalence issue is a correctness gap, not self-reference.

full rationale

The paper's central claim is a Γ-convergence proof: discrete minimisers of the DG energies converge to minimisers of the relaxed quasiconvex-envelope problem. The derivation chain does not reduce to its own inputs. The continuous reconstruction estimate Lemma 2.2 is a generalisation of Karakashian–Pascal [28], with the proof sketched but not assuming the desired convergence; the compactness argument uses the external Buffa–Ortner embeddings; the relaxation step uses Dacorogna [21]. The authors' prior work [27] is cited for a DG Poincaré inequality and for the recovery-sequence template, but those are auxiliary established results, not the quasiconvex conclusion being proved. No parameter is fitted to the target limit: α is only required to be sufficiently large, and no numerical data are used as input to the convergence theorems. The main legitimate concern is Remark 3.1: the penalty (3.6) can be undefined in real arithmetic for admissible W with a sufficiently negative lower bound, and the claimed comparability with (3.8) is not justified under the stated assumptions. This is a correctness/completeness gap in the proof's coverage of the stated scheme, not a circular reduction—the proof does not define the target in terms of itself, and the theorems would still constitute an independent argument for the (3.8)-type functional. Hence the paper has no significant circularity, only minor benign self-citation and a non-circular technical gap.

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

The paper introduces no fitted free parameters: alpha is an existence threshold, not a fitted value. The external inputs are standard regularity and growth assumptions on W, plus classical results on compact embeddings and relaxation. The new penalty (3.6) is a construction within the method, not an invented physical entity.

assumptions (5)
  • domain assumption W in C^1 satisfies growth and coercivity (3.2): -1+|xi|^p lesssim W(xi) lesssim 1+|xi|^p for p>1.
    Central assumption on the energy density, invoked throughout Section 3 for compactness and for the quasiconvex envelope properties.
  • domain assumption W satisfies the (p-1)-Lipschitz continuity (3.3): |W(xi1)-W(xi2)| lesssim (1+|xi1|^{p-1}+|xi2|^{p-1})|xi1-xi2|.
    Used to control the nonlinear differences in Lemmas 3.2 and 3.3; the printed formula repeats |xi1|^{p-1}, a typo for the symmetric form with |xi2|^{p-1}.
  • standard math Buffa-Ortner compact embedding and trace lemmas [11, Theorem 5.2, Lemma 8] for broken Sobolev spaces W^{1,p}(Omega,T_h).
    Invoked in Lemma 3.1 to extract L^p convergent subsequences and L^p(partial Omega) convergent traces; external result assumed valid.
  • standard math Dacorogna's relaxation theorem [21, Theorem 9.1]: L^p approximation sequences u_k to u with E(u_k) to E^{qc}(u).
    Used in Lemma 3.3 to construct recovery sequences for arbitrary u in A(Omega).
  • standard math Lifting operator bound (3.35) from [11,22] for the discrete-gradient method.
    Used in Section 3.2 to control the difference between the discrete gradient and the reconstructed gradient.

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Pith. "Pith review of Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems." pith.science (2026). https://pith.science/paper/GFKO4MUI

@misc{pith2026250111944,
  author       = {Pith},
  title        = {Pith review of: Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFKO4MUI}},
  note         = {Machine review of arXiv:2501.11944}
}
read the original abstract

In this work, we establish that discontinuous Galerkin methods are capable of producing reliable approximations for a broad class of nonlinear variational problems. In particular, we demonstrate that these schemes provide essential flexibility by removing inter-element continuity while also guaranteeing convergent approximations in the quasiconvex case. Notably, quasiconvexity is the weakest form of convexity pertinent to elasticity. Furthermore, we show that in the non-convex case discrete minimisers converge to minimisers of the relaxed problem. In this case, the minimisation problem corresponds to the energy defined by the quasiconvex envelope of the original energy. Our approach covers all discontinuous Galerkin formulations known to converge for convex energies. This work addresses an open challenge in the vectorial calculus of variations: developing and rigorously justifying numerical schemes capable of reliably approximating nonlinear energy minimization problems with potentially singular solutions, which are frequently encountered in materials science.

Figures

Figures reproduced from arXiv: 2501.11944 by the authors.

Figure 1
Figure 1. Total potential energy (Vertical axis) for the ho￾mogeneous deformation y0(x) = F0x (dashed black line), for the computed minimiser with respect to the stabilisation pa￾rameter α employing the penalty of eq. 4.2 (blue squares) and the penalty of eq. 3.6 (orange circles). We have noticed that when p = 8 roundoff errors affects the discrete minimi￾sation process. Note that the second term of the proposed penalty, eq.(… view at source ↗
Figure 2
Figure 2. Density (1/ det ∇y) in the deformed configura￾tion under uniaxial compression (10% strain). Employing the penalty of eq. (4.2), the density for the computed minimisers are illustrated: (a) when α = 20 and (b) when α = 160 (see blue squares of Figs. 3a and 3b at α = 20, 160). (c): Com￾puted solution when the proposed penalty of eq.(3.6) is used for α = 20. (d) Density of the exact homogeneous minimiser y0 [PITH_FULL… view at source ↗
Figure 3
Figure 3. Comparing the accuracy of the two proposed sta￾bilisation penalty terms (3.6) (circles) and (4.2) (squares) for the polyconvex W(F) = | det F| 2 strain energy under 10% uniaxial compression. Horizontal axis: values of the penalty parameter α. Vertical axes: |yh − y0|L1(Ω) and |yh − y0|W1,1(Ω) errors for various mesh resolutions, specif￾ically for 1024 (blue), 2034 (green) and 4096 (orange) tri￾angles, where yh denot… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Imposing the homogeneous boundary conditions y0(x) = (0.5R1V + 0.5I)x, minimisers are computed varying the mesh resolution (horizontal axis). Blue curve denotes the total potential energy (scale of left vertical axis) and red dashed curve is the error ||yh−y0||L2(Ω) (s…
Figure 5
Figure 5. Figure 5: The maximum eigenvalue (λmax) in the deformed configuration of the computed minimiser for each triangle. The wells I, V have the eigenvalues (1, 1) and 0.9, 1.09871212 respectively. In blue regions λmax = 1 which indicates that well I is attained up to a rotation. Simi…
Figure 6
Figure 6. Figure 6: The maximum eigenvalue (λmax) of the discrete minimiser for each triangle expressed in the reference con￾figuration. The wells I, V have the eigenvalues (1, 1) and 0.9, 1.09871212 respectively. In blue regions λmax = 1 which indicates the well I appears up to a rotatio…
Figure 7
Figure 7. Figure 7: Displacement of the computed minising sequence varying the mesh resolution. (a): Pointwise values of |u| = √ u · u, u : Ω → R 2 is the displacement vector. For 200, 800, and 32768 triangles the values of |u| along the diagonal of the domain (white line in (a)) are illu…

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Pith tools

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