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

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