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

We consider the numerical approximation of acoustic wave propagation problems by mixed BDMk+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {BDM}_{k+1}$$\end{document}–Pk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {P}_k$$\end{document} finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure 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 also apply for non-convex domains and non-uniform meshes.