M. Parvizi:

"Hierarchical methods in the discretization of elliptic problems";

Supervisor, Reviewer: J. Melenk, S. Beuchler, S. Börm; Institut für Analysis und Scientific Computing, 2021; oral examination: 2021-06-15.

In this thesis, we analyze the following multilevel aspects in elliptic boundary value problems:

- Multilevel representation of Besov norms and application to preconditioning of the fractional Laplacian.

- Use of hierarchical matrices (H-matrices) for the coupling of Finite- and Boundary Element Methods (FEM-BEM couplings).

- H-matrix approximability of inverses of matrices corresponding to the discretization of the time-harmonic Maxwell equations using Finite Element Method (FEM).

We show that locally L2()-stable operators mapping into spaces of continuous piecewise polynomial set on shape regular meshes with certain approximation properties in L2()are stable mappings

H3=2() ! B3=2 2;1(), where Hs() and Bs 2;q() are Sobolev and Besov

spaces. The classical Scott-Zhang type operators are included in the setting. Interpolation gives stability B3 =2 2;q () ! B3 =2

2;q (), 2 (0; 1); q 2 [1;1]. An analogous result allows

for spaces of discontinuous piecewise polynomials: locally L2-stable operators such as the elementwise L2-projection are stable B =2

2;q () ! B =2 2;q (), 2 (0; 1); q 2 [1;1].

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 B3 =2

2;q (), 2 (0; 1); q 2 [1;1].

As an application, we present a multilevel diagonal preconditioner for the integral fractional Laplacian ( )s for s 2 (0; 1) on locally re ned meshes. This preconditioner is shown to lead to uniformly bounded condition numbers.

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 scaler transmission problem.

H-matrices are a class of matrices that consists of blockwise low-rank matrices of rank r where the blocks are organized in a tree TI so that the memory requirement is typically O(Nr depth(TI)), 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 the chosen format.

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