Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions

Springer Science and Business Media LLC - Tập 72 - Trang 313-348 - 1996
Maksymilian Dryja1, Marcus V. Sarkis2, Olof B. Widlund3
1Department of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland e-mail: [email protected] , , PL
2Department of Computer Science, University of Colorado, Boulder, CO 80309-0430, USA e-mail: [email protected] , , US
3Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA e-mail: [email protected] , , US

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.