Comparative study of continuously differentiable Galerkin time discretizations for the wave equation

Abstract

In this work we compare numerically two families of space-time finite element approximation schemes for wave equations. Continuously differentiable in time discrete solutions are provided by the schemes. The first family of schemes is based on a computationally cheap post-processing of the continuous in space and time finite element approximation of the wave equation. The second family combines the variational approximation in time by finite element techniques with the concepts of collocation methods. Both families show equal order of convergence. The corresponding error constants are reviewed here.

$C^1$-conforming variational discretization of the biharmonic wave equation

“…The conceptual basis of the families of approximations to the wave equation is the establishment of a connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex algebraic systems. Further numerical studies for the wave equation can be found in [11,5]. For the application of the Galerkin-collocation to mathematical models of fluid flow and systems of ordinary differential equations we refer to [4,15,16].…”