Local discontinuous Galerkin FEM for convex minimization

TL;DR

This paper introduces a refined duality-based analysis of local discontinuous Galerkin methods for convex minimization, achieving improved convergence rates and effective error control.

Contribution

It develops a novel duality framework for dG schemes that enhances convergence rates and provides reliable a posteriori error estimates for convex minimization.

Findings

Improved a priori convergence rates for dG schemes in convex minimization.

Effective a posteriori error control enabling adaptive mesh refinement.

Numerical benchmarks demonstrate superior convergence with adaptive refinement.

Abstract

The heart of the a priori and a posteriori error control in convex minimization problems is the sharp control of the differences of discrete and exact minimal energy. Conforming finite element discretizations for p-Laplace type minimization problems provide upper bounds of the energy difference with optimal convergence rates. Even for smooth solutions, known convergence rates for higher-order non-conforming finite element discretizations for the same problem class with $2 < p < \infty$, however, are exclusively suboptimal. Thus the popular a posteriori error control within the two-energy principle, that generalize hyper-circle identities, appears unbalanced. The innovative point of departure in a refined analysis of two discontinuous Galerkin (dG) schemes exploits duality relations between a discrete primal and a semi-discrete dual problem. The infinite-dimensional dual problem leads to a tiny duality gap that even vanishes for polynomial low-order terms. For a class of degenerated convex minimization problems with two-sided $p$ growth, the novel duality provides improved a priori convergence rates for the error in the minimal energies. This closes the misfit of convergence rates for the conforming and nonconforming schemes at least for the local discontinuous Galerkin schemes at hand. The motivating two-energy principle and some post-processing for a Raviart-Thomas dual variable provides an a posteriori error control, that also may drive adaptive mesh-refining. Computational benchmarks provide striking numerical evidence for improved convergence rates of the adaptive beyond uniform mesh-refining.