Robust Algebraic Multigrid Methods in Magnetic Shielding Problems

Stefan Reitzinger

Johannes Kepler University
Institut fur Mathematik 
Altenbergerstrasse 69
A-4040 Linz
Austria                     

Abstract

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 H<sup>1</sup> is presented.

To achieve an efficient and robust solution strategy the algebraic multigrid
method of Ruge and Stueben 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<sup>-3</sup>.
Additionally parameter jumps of order 10<sup>-6</sup> to 10<sup>+6</sup> 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.