APOWA

Transient wave propagation problems are ubiquitous in a large range of applications and are often modeled as linear hyperbolic partial differential equations (PDEs). In complex geometries, finite element and discontinuous Galerkin methods are popular approaches to compute approximate solutions. However, although there is a vast literature concerning the convergence analysis of such schemes on uniform meshes, the development of explicit error bounds and solution-adapted locally refined meshes is missing. In practice, this means (a) that it is hard to reliably predict the actual error associated with a given mesh and (b) that most of the computational resources are wasted in simulations, since a significant amount of mesh elements could be either removed or enlarged without sacrificing accuracy. The goal of this project is to address these two issues and provide guaranteed error bounds and optimally adapted meshes for transient wave propagation problems. This will be achieved by using a posteriori error estimation and adaptive mesh refinements. The combined use of a posteriori error estimators and adaptive mesh refinements provides a very convincing solution to the above-mentioned issues for elliptic PDEs, and the theory has more recently been extended to parabolic PDEs. Nevertheless, hyperbolic PDEs have not been treated in the past, and this project aims to bridge the theories of a posteriori error estimation and adaptive mesh refinements to PDEs modelling transient wave propagation problems. This approach is challenging since as compared to elliptic and parabolic PDEs, hyperbolic PDEs are not inf-sup stable in natural Sobolev norms. The starting point of this endeavor is a recent breakthrough developed by the principal investigator that bypasses this lack of inf-sup stability and opens a path towards the development of a posteriori error estimators and optimally convergent adaptive schemes that will be at the heart of this project.