My main field of research is the discretization and numerical solution of partial differential equations (PDEs). I develop finite element and discontinuous Galerkin approaches for various elliptic and hyperbolic systems of PDEs with a focus on non-conforming meshes and parallel solvers.
In cooperation with geophysicists I am working on science applications, such as the simulation of thermal convection in the earth's mantle, and global scale seismic wave propagation, where high performance computing is essential to achieve the resolution necessary to capture important physical phenomena. A related field of interest are Newton-Krylov methods for ill-posed inverse problems with applications in full waveform inversion for seismology and uncertainty quantification in the simulation of ice sheets.
Many physical systems exhibit multiscale phenomena that lend themselves to simulations using adaptive meshes. On large parallel computers however, and in particular for problems that require frequent refinement, coarsening, and repartitioning of the mesh, it is a challenge to perform such mesh operations efficiently. I have worked extensively on the forest-of-octrees approach and designed scalable algorithms for parallel AMR. These are implemented in the p4est software library that is available under the GNU GPL version 2 (or later). p4est has been used as parallel mesh infrastructure for a number of large-scale simulation projects; see my publications page for more details. Another software project related to parallel adaptive mesh refinement that I am working on is ForestClaw.