Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions
Tóm tắt
Multilevel Schwarz methods are developed for a
conforming finite element approximation of second order elliptic problems. We
focus on problems in three dimensions with
possibly large jumps in the coefficients across the
interface separating the subregions. We establish
a condition number estimate for the iterative operator, which is
independent of the coefficients, and grows at most as the square
of the number of levels. We also characterize a class of distributions
of the coefficients,
called quasi-monotone, for which the weighted
$L^{2}$
-projection is
stable and for which we can use the standard piecewise
linear functions as a coarse space. In this case,
we obtain optimal methods, i.e. bounds which are independent of the number
of levels and subregions. We also design and analyze multilevel
methods with new coarse spaces
given by simple explicit formulas. We consider nonuniform meshes
and conclude by an analysis of multilevel iterative substructuring methods.