M. Faustmann:

"Approximation to inverse finite element- and boundary element matrices with hierarchical matrices";

Supervisor, Reviewer: J. Melenk, W. Hackbusch; Institut für Analysis und Scientific Computing, 2021; oral examination: 2015-10-14.

This thesis is concerned with the existence of approximations to the inverses of matrices that are obtained from finite element or boundary element discretizations of some partial differential equations. The approximation is performed in the format of hierarchical matrices. The concept of hierarchical matrices permits to store as well as to apply some (approximate) arithmetic operations in logarithmic linear complexity. The inverses of finite element and boundary element matrices are dense in general, which leads to slow computations and large amounts of memory consumption. Therefore, the approximability of these matrices by hierarchical matrices is desirable. In this thesis, we show that approximations to inverses of finite element matrices for second order elliptic partial differential equations with mixed boundary conditions and for the Lam'e equation exist, and the error converges exponentially in the block-rank of the approximation. Our results generalize the known theoretical results for elliptic partial differential equations and the Lame system with Dirichlet boundary conditions to more general boundary conditions. Moreover, the existence of approximations with arbitrary accuracy is proven, whereas the previous results only achieved accuracy up to the discretization error. For the inverses of boundary element matrices the applicability of hierarchical matrices has only been observed numerically. One of the main results of this work is the proof of existence of an approximation to the inverse matrices corresponding to discretizations of the single-layer and the hypersingular integral operator. These results give a mathematical foundation to the numerically observed success of hierarchical matrices for the approximation of inverse matrices. In contrast to the known results in the literature, we work in a fully discrete setting, which leads to approximations of arbitrary accuracy. A main ingredient for our purpose is the proof of a discrete interior regularity result similar to the classical Caccioppoli inequality. Besides the approximations to inverse matrices, we are also concerned with the existence of approximative Cholesky- and $LU$-decompositions for the above mentioned finite element and boundary element discretizations. By approximating certain Schur complements by some blockwise low rank factorizations, we are able to prove the existence of approximate Cholesky- and LU-decompositions in the format of hierarchical matrices. For example, these factorizations can be used for black-box preconditioning in iterative solvers. A series of numerical examples is given, which confirm the theoretical results.

hierarchical matrices/rank-structured matrices/finite element method/boundary element method

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