Super-convergence and post-processing for mixed finite element approximations of the wave equation
For researchers in numerical analysis and computational wave propagation, this work provides a rigorous theoretical foundation and practical post-processing method to enhance solution accuracy, though it is an incremental extension of existing techniques from elliptic to hyperbolic problems.
The paper establishes optimal convergence and super-convergence properties for mixed finite element approximations of the wave equation, and proposes a post-processing strategy to improve pressure accuracy. The results extend known techniques from elliptic problems to hyperbolic problems, with proofs applicable to non-convex domains and non-uniform meshes.
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.