Finite element approximation of the acoustic wave equation: error control and mesh adaptation

We present an approach to solving the acoustic wave equation by adaptive finite element methods. Using a global duality argument and Galerkin orthogonality, we obtain a residual-based error representation with respect to an arbitrary functional of the solution. This results in numerically evaluatable error estimates which are used for mesh refinement. In this way, very economical and highly localized space-time meshes can be generated which are tailored to the efficient computation of the quantity of interest. We demonstrate the performance and some of the mechanisms acting in our approach by numerical examples.