Space-time finite elements for the wave propagation problem

Summary

The wave propagation problem has been traditionally approximated by finite differences, or by the Galerkin method. The latter, in a finite element contest, implies the approximation across the space variables by finite elements, and the integration of the resulting sistem of differential equations on time by finite differences. The weak, variational formulation is used only for lowering the space derivatives. The present work attempts at discretising and solving a weak formulation on space and time due to Lions [1]. The finite elements extend simultaneously on space and time. The resulting set of linear equations, however, usually outgrows by an order of magnitude in matrix size and bandwidth, the corresponding dimensions with a Galerkin method. That consideration has led to an early dismissal of the space-time approach.

Assuming constant time step, the matrix takes a recursive structure, which allows for reduction of the solution of a set of linear equations of Galerkin size. A simple example of one dimensional wave propagation is solved by space-time finite elements and the results are compared with finite differences and with the Galerkin method.