MULTILEVEL SCHWARZ METHODS FOR ELLIPTIC PROBLEMS WITH DISCONTINUOUS COEFFICIENTS IN 3 DIMENSIONS

Multilevel Schwarz methods are developed for a conforming finite eleme nt approximation of second order elliptic problems, We focus on proble ms in three dimensions with possibly large jumps in the coefficients a cross the interface separating the subregions. We establish a conditio n number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of lev els, We also characterize a class of distributions of the coefficients , called quasi-monotone, for which the weighted L(2)-projection is sta ble and for which we can use the standard piecewise linear functions a s 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.