Discontinuous Galerkin for the wave equation: a simplified a priori error analysis

TL;DR

This paper presents a simplified a priori error analysis for discontinuous Galerkin methods applied to the wave equation, achieving optimal order error estimates with minimal regularity assumptions and verifying results through numerical experiments.

Contribution

It provides a straightforward a priori error analysis for DG methods on wave equations, with minimal regularity requirements and uniform norm error estimates.

Findings

Optimal order error estimates achieved

Minimal regularity needed for solutions

Numerical experiments confirm theoretical results

Abstract

Standard discontinuous Galerkin methods, based on piecewise polynomials of degree $ \qq=0,1$, are considered for temporal semi-discretization for second order hyperbolic equations. The main goal of this paper is to present a simple and straightforward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved. To this end, energy identities and stability estimates of bthe discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used, to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical results.