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 quasimonotone, for which the weighted L²-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.

Additive multilevel Schwarz methods are developed for conforming finite element approximations 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 condition number estimates for the iterative operators, which are independent of the coefficients, and grow at most as the square of the number of levels `. For the multiplicative versions, such as the V-cycle multigrid methods using Gauss Seidel and damped Jacobi smoothers, we obtain a rate of convergence bounded from above by 1 \Gamma C ` \Gamma2 . We also characterize a class of distributions of values of the coefficients, called quasi-monotone, for which the error of the weighted L 2 -projection is stable and for which we can use the standard piecewise linear functions as a coarse space and obtain condition number estimates independent of the number of levels, subregions, and the coefficients. We also des...

for elliptic problems with discontinuous coefficients in three dimensions