A Discontinuous Ritz Method for a Class of Calculus of Variations Problems

TL;DR

This paper introduces the discontinuous Ritz (DR) method, a novel approach inspired by discontinuous Galerkin techniques, for approximating calculus of variations problems by minimizing a discrete energy functional.

Contribution

It develops a new DR method that incorporates DG finite element derivatives and proves its convergence for complex energy functionals.

Findings

The DR method converges for a class of calculus of variations problems.

Numerical tests on the p-Laplace problem demonstrate the method's effectiveness.

DG-FE derivatives exhibit a crucial compactness property.

Abstract

This paper develops an analogue (or counterpart) to discontinuous Galerkin (DG) methods for approximating a general class of calculus of variations problems. The proposed method, called the discontinuous Ritz (DR) method, constructs a numerical solution by minimizing a discrete energy over DG function spaces. The discrete energy includes standard penalization terms as well as the DG finite element (DG-FE) numerical derivatives developed recently by Feng, Lewis, and Neilan in [Feng2013]. It is proved that the proposed DR method converges and that the DG-FE numerical derivatives exhibit a compactness property which is desirable and crucial for applying the proposed DR method to problems with more complex energy functionals. Numerical tests are provided on the classical $p$-Laplace problem to gauge the performance of the proposed DR method.