Variational Space–Time Methods for the Wave Equation

In this work we present some new variational space–time discretisations for the scalar-valued acoustic wave equation as a prototype model for the vector-valued elastic wave equation. The second-order hyperbolic equation is rewritten as a first-order in time system of equations for the displacement and velocity field. For the discretisation in time we apply continuous Galerkin–Petrov and discontinuous Galerkin methods, and for the discretisation in space we apply the symmetric interior penalty discontinuous Galerkin method. The resulting algebraic system of equations exhibits a block structure. First, it is simplified by some calculations to a linear system for one of the variables and a vector update for the other variable. Using the block diagonal structure of the mass matrix from the discontinuous Galerkin discretisation in space, the reduced system can be condensed further such that the overall linear system can be solved efficiently. The convergence behaviour of the presented schemes is studied carefully by numerical experiments. Moreover, the performance and stability properties of the schemes are illustrated by a more sophisticated problem with complex wave propagation phenomena in heterogeneous media.