Super-convergence and post-processing for mixed finite element approximations of the wave equation

TL;DR

This paper analyzes mixed finite element methods for wave equations, demonstrating optimal convergence, super-convergence of pressure, and proposing a post-processing technique to enhance pressure approximation, applicable to fully discrete schemes on unstructured meshes.

Contribution

It extends super-convergence and post-processing techniques from elliptic to hyperbolic wave problems using mixed finite elements, without relying on duality or inverse inequalities.

Findings

Optimal convergence of velocity approximation

Super-convergence of pressure by one order

Effective post-processing method for improved pressure accuracy

Abstract

We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank-Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore apply also for non-convex domains and non-uniform meshes.