# Finite Element Heterogeneous Multiscale Method for the Wave Equation

**Marcus J. Grote
**

Department of Mathematics

University of Basel

Basel, Switzerland

### Abstract:

Following previous work of Abdulle, E, and Engquist (2003, 2005),
we propose a finite element heterogeneous multiscale
method (HMM) for the simulation of time dependent waves propagating through
a medium with rapidly varying sound speed. Here the fine scales for the
problem are accounted for in the stiffness matrix at the coarse scale
by solving inside each macro-element a few micro-problems
each of fixed size independent of the finest scale. A priori error estimates
of the fully discrete method, which include the discretization
errors from the macro- and the micro-problems, are derived.
Numerical examples illustrate the theoretical convergence rates
and the usefulness of this approach.