In this project, the investigator and his colleagues will develop, analyze, and implement quasi-optimal adaptive hybridized Discontinuous Galerkin methods for acoustic and electromagnetic scattering problems as described by the Helmholtz equation and the time-harmonic Maxwell equation on bounded domains in two and three dimensional domains using general local bases. These methods feature traditional piecewise polynomial bases as well as special purpose wave elements, and multilevel preconditioned iterative solvers on adaptively refined simplicial, quadrilateral, and hexahedral meshes. The adaptive mesh refinement is driven by residual-type a posteriori error estimators. In particular, the team will prove convergence of the adaptive solution process as well as its quasi-optimality in terms of the computational complexity with respect to a properly specified approximation class. High quality software implementations will be used to test the algorithms and inform the analysis.
Advances in the state of the art in numerical methods for performing acoustic and electromagnetic simulations have potentially high impact in several applications. As an example more accurate simulation of acoustic wave propagation can be used to considerably enhance resolution of medical ultra sound scans, and thus expand the reach of non-invasive diagnostics. The emphasis of this project is the development of novel numerical methods that compute provably accurate approximations of physical acoustics and electromagnetics phenomena.
