Superconvergence of Discontinuous Galerkin methods for Elliptic Boundary Value Problems

TL;DR

This paper provides a unified theoretical analysis of superconvergence in mixed discontinuous Galerkin methods for elliptic boundary value problems, demonstrating improved accuracy through postprocessing and validating results with numerical experiments.

Contribution

It offers the first unified analysis of superconvergence for a broad class of mixed DG methods applied to elliptic problems, including Poisson and elasticity.

Findings

Superconvergence occurs for mixed DG methods with polynomial degree k.

Postprocessing schemes enhance displacement accuracy by order min(k+1, 2).

Numerical experiments confirm theoretical superconvergence results.

Abstract

In this paper, we present a unified analysis of the superconvergence property for a large class of mixed discontinuous Galerkin methods. This analysis applies to both the Poisson equation and linear elasticity problems with symmetric stress formulations. Based on this result, some locally postprocess schemes are employed to improve the accuracy of displacement by order min(k+1, 2) if polynomials of degree k are employed for displacement. Some numerical experiments are carried out to validate the theoretical results.