Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems

TL;DR

This work shows that discontinuous Galerkin methods can reliably approximate nonlinear, vectorial variational problems with quasiconvex energies, and that DG minimisers converge to minimisers of the relaxed energy $\mathcal{E}^{qc}$ defined via the quasiconvex envelope $W^{qc}$. By proving $\Gamma$-convergence of the discrete energies to $\mathcal{E}^{qc}$ and analyzing two DG formulations (including one based on discrete gradients), the authors provide a rigorous foundation for numerical schemes that remain stable and convergent even in nonconvex elasticity problems with potential singularities. The results are supported by computational experiments on polyconvex and multi-well energies, illustrating convergence toward the relaxed minimiser, the formation of microstructures, and the potential to approximate $W^{qc}$ pointwise from discrete energies. Together, these findings address a long-standing challenge in the vectorial calculus of variations by delivering flexible, provably convergent DG approaches for nonlinear energy minimisation in materials science.

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 (7)

• Figure 1: Total potential energy (Vertical axis) for the homogeneous deformation $y_0(x) = F_0 x$ (dashed black line), for the computed minimiser with respect to the stabilisation parameter $\alpha$ employing the penalty of eq. \ref{['eq:convexpen']} (blue squares) and the penalty of eq. \ref{['eq:penalty']} (orange circles).
• Figure 2: Density ($1/\det \nabla{y}$) in the deformed configuration under uniaxial compression (10% strain). Employing the penalty of eq. \ref{['eq:convexpen']}, the density for the computed minimisers are illustrated: \ref{['fig:convex_a1']} when $\alpha=20$ and \ref{['fig:convex_a4']} when $\alpha =160$ (see blue squares of Figs. \ref{['fig:fn1c']} and \ref{['fig:fn1d']} at $\alpha =20, 160$). \ref{['fig:quasiconvex_a1']}: Computed solution when the proposed penalty of eq.\ref{['eq:penalty']} is used for $\alpha=20$. \ref{['fig:exact']} Density of the exact homogeneous minimiser $y_0$.
• Figure 3: Comparing the accuracy of the two proposed stabilisation penalty terms \ref{['eq:penalty']} (circles) and \ref{['eq:convexpen']} (squares) for the polyconvex $W(F) = |\det F|^2$ strain energy under 10% uniaxial compression. Horizontal axis: values of the penalty parameter $\alpha$. Vertical axes: $|y_h - y_0|_{L^1(\Omega)}$ and $|y_h - y_0|_{W^{1,1}(\Omega)}$ errors for various mesh resolutions, specifically for 1024 (blue), 2034 (green) and 4096 (orange) triangles, where $y_h$ denote the numerical solutions and $y_0$ the homogeneous minimiser. In \ref{['fig:fn1c']} circular error belongs in the range of $10^{-8}$ and $10^{-9}$, while in \ref{['fig:fn1d']} circular errors are of the order $10^{-6}$.
• Figure 4: Imposing the homogeneous boundary conditions $y_0(x) = (0.5R_1 V +0.5 I)x$, 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 $||y_h - y_0||_{L^2(\Omega)}$ (scale of right vertical axis) where $y_h$ denote the computed minimiser.
• Figure 5: The maximum eigenvalue ($\lambda_{max}$) 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 $\lambda_{max} =1$ which indicates that well $I$ is attained up to a rotation. Similarly, red color, $\lambda_{max} =1.09871212$, corresponds to $R_i V$ up to a rotation, $i=1,2$.

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.1: Poincaré inequality for DG spaces.
• proof
• Remark 3.1
• Theorem 3.1
• Lemma 3.1
• proof