| Popis: |
In this thesis, we prove that the inverse of a certain type of Gram matrix can be approximated well from the class of hierarchical matrices. The entries of a Gram matrix are determined by a bilinear form on a suitable function space and by a finite set of basis functions. Such matrices appear frequently in the context of Galerkin discretizations of partial differential equations and many related problems in physics, engineering and applied mathematics. As for the function spaces, we are mainly concerned with the usual Sobolev spaces of integer order and the bilinear forms under consideration are variants of the inherent inner products on these spaces. An important prerequisite for the analysis is the validity of a discrete Caccioppoli inequality, which we derive in a finite element setting and a radial basisfunction setting. The assumptions on the basis functions that make up the Gram matrix are very mild and do not incorporate locality. In fact, we only require some form of locality for the corresponding dual basis. The question of low-cost approximability of inverse Gram matrices is certainly not new. In more than two decades of research, different approaches have been made to answer thisquestion (e.g., [Bebendorf & Hackbusch 2003], [Börm 2010], [Faustmann et al. 2015]). The goal of this work is to unify these ideas and rephrase them in a more abstract framework, which can be applied to a larger class of problems. In particular, we can treat mesh-based and mesh-less problems, locally refined meshes/point clouds in arbitrary space dimensions, rough PDE coefficients, non-polygonal computational domains and non-local basis functions. This thesis is based on the papers [Angleitner, Faustmann, Melenk 2021a], [Angleitner, Faustmann, Melenk 2022] and [Angleitner, Faustmann, Melenk 2022], which were composed as part of the author’s doctoral studies at Technische Universität Wien in collaboration with Dr. Markus Faustmann and Univ.-Prof. Jens Markus Melenk, PhD. |
