Hierarchical techniques in the discretization of elliptic boundary value problems

Abstract

In this thesis, we analyze the following multilevel aspects in elliptic boundary value problems:1) multievel representations of Besov norms and application to precondition of the fractional Laplacian.2) Use of hierarchical matrices (H-matrices) for the couping finite and boundary element methods (FEM-BEM coupling)3) H-matrix approximabiity of inverses of matrices correcponding to the discretization of the time-harmonic Maxwell equations using the finite element method (FEM). We show that locally L^2(\Omega)-stable operators mapping into spaces of continuous piecewise polynomials on shape regular meshes with certain approximation properties in L^2(\Omega) are stable mappings H^{3/2}(\Omega) \rightarrow B^{3/2}_{2,\infty}(\Omega), where H^s(\Omega) and B^s_{2,q}(\Omega) are Sobolev and Besov spaces. The classical Scott-Zhang type operators are included in the setting. Interpolation gives stability B^{\theta/2}_{2,q}(\Omega) \rightarrow B^{\theta/2}_{2,q}(\Omega) for \theta \in (0,1) and q \in [1,\infty].. An analogous result holds for spaces of discontinuous piecewise polynomials: locally L^2 -stable operators such as the elementwise L^2-projection are stable B^{\theta/2}_{2,q}(\Omega) \rightarrow B^{\theta/2}_{2,q}(\Omega), \theta \in (0,1), q \in [1,\infty]. For spaces of piecewise polynomials on adaptively refined meshes generated by Newest Vertex Bisection (NVB), we construct a multilevel decomposition with norm equivalence in the Besov space B^{3\theta/2}_{2,q}(\Omega), \theta \in (0,1), q \in [1,\infty]. As an application, we present a multilevel diagonal preconditioner for the integral fractional Laplacian (-\Delta)^s for s \in (0,1) on locally refined meshes. This preconditioner is shown to lead to uniformly bounded condition number.This work is also concerned with approximation results for the inverses of stiffness matrices corresponding to the FEM and FEM-BEM discretizations in the H-matrix format for the time-harmonic Maxwell equation and a scalar transmission problem. H-matrices are a class of matrices that consist of blockwise low-rank matrices of rank r where the blocks are organized in a tree T so that the memory requirement is typicallyO(N r depth(T )), where N is the problem size. A basic question in connection with the H-matrix arithmetic is whether matrices and their inverses can be represented well in thec hosen format. We consider three different methods for the coupling of the FEM and the BEM, namely, the Bielak-MacCamy coupling, the symmetric coupling, and the Johnson-Nédélec coupling. For the lowest order Galerkin discretization of each of these coupling techniques we prove the existence of root exponentially convergent H-matrix approximants to the inverse matrices. We also show that the inverse of the stiffness matrix corresponding to the time-harmonic Maxwell equations with perfectly conducting boundary conditions can be approximated in the format of H-matrices, at a root exponential rate in the block rank. In order to prove these H-matrix approximability results, we provide interior regularity results known as Caccioppoli estimates for the discrete problems. For the FEM-BEM coupling, the Caccioppoli inequality allows for control of functions and induced potentials in stronger norms by weaker norms, if certain orthogonality conditions are satisfied. For the Maxwell equations, the Caccioppoli estimate takes the form of control of the H(curl)-norm by the L^2 -norm.