A class of Discontinuous Galerkin methods for nonlinear variational problems

TL;DR

This paper introduces a new class of Discontinuous Galerkin methods for nonlinear variational problems, demonstrating convergence of discrete solutions to the continuous minimizer through theoretical analysis and numerical validation.

Contribution

It proposes element-wise nonconforming finite element methods with penalty terms for nonlinear problems, establishing convergence via $Gamma$-convergence and providing numerical evidence.

Findings

Discrete minimizers converge to the continuous solution as mesh size decreases.

The proposed methods are feasible and effective for nonlinear variational problems.

Numerical examples confirm theoretical convergence results.

Abstract

In the context of Discontinuous Galerkin methods, we study approximations of nonlinear variational problems associated with convex energies. We propose element-wise nonconforming finite element methods to discretize the continuous minimisation problem. Using $\Gamma$-convergence arguments we show that the discrete minimisers converge to the unique minimiser of the continuous problem as the mesh parameter tends to zero, under the additional contribution of appropriately defined penalty terms at the level of the discrete energies. We finally substantiate the feasibility of our methods by numerical examples.