The aim of this diplom thesis is to provide a robust and efficient solver for large sparse and poor conditioned linear systems arising from the FE-method for elliptic scalar PDEs of second order.
For a counter example the problem of magnetic shielding is used. Therefore the Maxwell's equations for stationary objects are reduced to a scalar PDE of second order with appropriate boundary conditions.
In order to solve the equation by means of FEM, a discretization for micro scales is introduced. Especially long thin elements are suggested to keep the number of unknowns small in areas of micro structures. Constructively a finite element analysis is carried out where also a convergence result of the FE-solution in H1 is presented.
To achieve an efficient and robust solution strategy the algebraic multigrid method of Ruge and Stüben is introduced. Additionally three different areas of application are presented for this AMG method, i.e. preconditioner, coarse grid solver for a full multigrid method, and black box solver.
Because this AMG method normally works well for M-matrices, a technique is presented to attain M-matrices, if the underlying linear system arises from an FE-discretization. The method to achieve the M-matrix property is based on the element matrices.
The algorithm was implemented as black box solver in the finite element package FEPP. Therein AMG was applied as preconditioner for the conjugate gradient method.
Some numerical experiments are presented, where long thin quadrilaterals are used with ratio of the longest and shortest side of 1 to 10-3. Additionally parameter jumps of order 10-6 to 10+6 are considered.
Concluding AMG has been proven, at least in a numerical way, to be an efficient and robust solver for magnetic shielding problems, if it is used as a preconditioner for the CG-method. If long thin quadrilaterals are used for discretization the modified preconditioner also behaves very robust.