Stefan Reitzinger

University of Linz, Austria

In this talk we will present 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. To achieve
an efficient and robust solution strategy the algebraic multigrid method of
Ruge and Stüben is used. 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. At least in a numerical
way, AMG has been proven to be an efficient and robust solver for magnetic
shielding problems, if it is used as a preconditioner for the CG-method. In
case of long thin quadrilaterals in the discretization the modified
preconditioner also behaves very robust.