Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems

TL;DR

This paper proves that discontinuous Galerkin methods can reliably approximate a wide range of nonlinear variational problems, including quasiconvex and non-convex cases, relevant in materials science.

Contribution

It establishes convergence of DG methods for quasiconvex and relaxed variational problems, addressing a key challenge in nonlinear energy minimization.

Findings

DG schemes provide reliable approximations for nonlinear variational problems.

Convergence is guaranteed in the quasiconvex case.

Discrete minimisers converge to relaxed problem minimisers.

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.