|Subject:||Algebraic Multigrid Methods for Large Scale Finite Element Equations|
|Author:||Dipl.-Ing. Stefan Reitzinger|
|Supervisor:||o.Univ.Prof. Dr. Ulrich Langer|
|External Referee:||o.Univ.Prof. Dr. R. H. W. Hoppe|
In these theses the algebraic multigrid method is considered for several dierent classes of matrices which arise from a finite element discretization of partial differential equations. In particular we consider system matrices that originate from a finite element discretization of some self-adjoint, elliptic partial differential equations of second order. Such matrices are typically symmetric and positive definite.
We are looking for an efficient, flexible and robust solution of the arising system of equations. The algebraic multigrid method is adequate to fulfill these requirements. A general concept to the construction of such multigrid methods is proposed. This concept provides an auxiliary matrix that allows us to construct optimal coarsening strategies, appropriate prolongation and smoothing operators. The auxiliary matrix mimics a geometric hierarchy of grids and comprehend operator- and mesh anisotropies. The key point consists in constructing a virtual finite element mesh and representing the degrees of freedom of the original system matrix on the virtual mesh appropriately. Based on this concept we derive algebraic multigrid methods for matrices stemming from Lagrange and Nédélec finite element discretizations. In addition, we present a necessary condition for the prolongation operator that ensures that the kernel of the system matrix (without essential boundary conditions) is resolved on a coarser level. The coarse grid operator is computed by the Galerkin method as usual. For the smoothing operator standard methods are taken, i.e., point-, or block Gauss-Seidel methods. One remedy for the often required M-matrix property of the system matrix is the so called element preconditioning technique for scalar self-adjoint equations. It turns out that this method can be seen as a special case of the general approach.
Apart from the discussed algebraic multigrid method approach a brief note on the setup phase is given along with the optimization for the setup phase for nonlinear, time-dependent and moving body problems. Furthermore, a parallel algebraic multigrid algorithm is presented, which is applicable to problems originating from Lagrange and Nédélec finite element discretizations.
All algebraic multigrid methods proposed in these theses are implemented in the software package PEBBLES. Finally, numerical studies are given for problems in science and engineering, which show the great potential of the proposed algebraic multigrid techniques.
This work has been supported by the Austrian Science Found - "Fonds zur Förderung der wissenschaftlichen Forschung" (FWF) - within the subproject F 1306 of the Special Research Program SFB F013 "Numerical and Symbolic Scientific Computing".
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