On the fast, efficient and robust numerical solution of partial differential equations in poro- and solid mechanics

University of Duisburg-Essen, Faculty of Mathematics, Thea-Leymann-Str. 9, 45127 Essen
Lymbery, Maria

This habilitation thesis consists of two main parts. The introductory part presents a summary that discusses briefly the content of the 11 included original research papers, their main motives, goals and relevance, and how they are related to each other. In the second part, these articles are exposed, grouped according to the problem classes to which they contribute, namely “Quasi-static poroelasticity” (6 papers), “Dynamic plate vibration” (1 paper) and “Scalar elliptic problems” (4 papers).

The first group of articles is mainly concerned with the stability analysis of coninuous and discrete mathematical models which fall into the category of quasi-static Biot and generalized Biot-Brinkman as well as multiple network poroelasticity theory (MPET) models. The proposed discretizations rely on structure preservation and exploit hybridized discontinuous Galerkin (DG) methods and mixed/hybrid-mixed formulations. Further, parameter-robust preconditioning techniques and uniformly convergent splitting schemes are proposed and analyzed, thus, providing efficient iterative solution methods for the arising large-scale systems of linear algebraic equations.

The second group, consisting of one paper, is devoted to the error analysis of a space-time variational discretization of the biharmonic wave equation. The time discretization is based on a combined Galerkin and collocation technique wheras space discretization employs the classical Bogner-Fox-Schmit element. Optimal order error estimates are proven.

The third group of papers focuses on the construction and analysis of optimal or nearly optimal multigrid algorithms for scalar elliptic problems. Here the main emphasis is on problems with strongly varying coefficients as they appear in the modeling of highly heterogeneous materials. As a main innovation, the concept of auxiliary space multigrid methods is introduced. In contrary to standard (variational) multigrid algorithms coarse-grid correction is replaced by an auxiliary space correction. The coarse-grid operator then appears as the exact Schur complement of the auxiliary matrix and defines an additive approximation of the Schur complement of the original system. A rigorous analysis of the two-level method is presented.


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