Wavenumber-Explicit -FEM Analysis for Maxwell's Equations with Impedance Boundary Conditions.

The time-harmonic Maxwell equations at high wavenumber in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly in and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of order on a mesh with mesh size is shown under the -explicit scale resolution condition that (a) / is sufficient small and (b) is bounded from below.

Approximating inverse FEM matrices on non-uniform meshes with -matrices.

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse -matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the -matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.

Runge-Kutta approximation for -semigroups in the graph norm with applications to time domain boundary integral equations.

We consider the approximation of an abstract evolution problem with inhomogeneous side constraint using -stable Runge-Kutta methods. We derive a priori estimates in norms other than the underlying Banach space. Most notably, we derive estimates in the graph norm of the generator.

Local convergence of the boundary element method on polyhedral domains.

The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm's integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local estimates in for Symm's integral equation and in for the hyper-singular equation. The local rate of convergence is limited by the local regularity of the sought solution and the sum of the rates given by the global regularity and additional regularity provided by the shift theorem for a dual problem.

FEM-BEM coupling for the large-body limit in micromagnetics.

We present and analyze a coupled finite element-boundary element method for a model in stationary micromagnetics. The finite element part is based on mixed conforming elements. For two- and three-dimensional settings, we show well-posedness of the discrete problem and present an error analysis for the case of lowest order elements.