Doubling the convergence rate by pre- and post-processing the finite element approximation for linear wave problems

Abstract: In this paper, a novel pre- and post-processing algorithm is presented that can double the convergence rate of finite element approximations for linear wave problems. In particular, it is shown that a $q$-step pre- and post-processing algorithm can improve the convergence rate of the finite element approximation from order $p+1$ to order $p+1+q$ in the $L^2$-norm and from order $p$ to order $p+q$ in the energy norm, in both cases up to a maximum of order $2p$, with $p$ the polynomial degree of the finite element space. The $q$-step pre- and post-processing algorithms only need to be applied once and require solving at most $q$ linear systems of equations. The biggest advantage of the proposed method compared to other post-processing methods is that it does not suffer from convergence rate loss when using unstructured meshes. Other advantages are that this new pre- and post-processing method is straightforward to implement, incorporates boundary conditions naturally, and does not lose accuracy near boundaries or strong inhomogeneities in the domain. Numerical examples illustrate the improved accuracy and higher convergence rates when using this method. In particular, they confirm that $2p$-order convergence rates in the energy norm are obtained, even when using unstructured meshes or when solving problems involving heterogeneous domains and curved boundaries.