ROBUST ITERATIVE METHODS FOR ELLIPTIC PROBLEMS WITH HIGHLY VARYING COEFFICIENTS IN THIN SUBSTRUCTURES

In this paper we introduce a class of robust multilevel interface solv ers for two-dimensional finite element discrete elliptic problems with highly varying coefficients corresponding to geometric decompositions by a tensor product of strongly non-uniform meshes, The global iterat ions convergence rate q < 1 is shown to be of the order q = 1 - O(log( -1/2) n) with respect to the number n of degrees of freedom on the sin gle subdomain boundaries, uniformly upon the coarse and fine mesh size s, jumps in the coefficients and aspect ratios of substructures. As th e first approach, we adapt the frequency filtering techniques [27] to construct robust smoothers on the highly non-uniform coarse grid. As a n alternative, a multilevel averaging procedure for successive coarse grid correction is proposed and analyzed. The resultant multilevel coa rse grid preconditioner is shown to have (in a two level case) the con dition number independent of the coarse mesh grading and jumps in the coefficients related to the coarsest refinement level. The proposed te chnique exhibited high serial and parallel performance in the skin dif fusion processes modelling [19] where the high dimensional coarse mesh problem inherits a strong geometrical and coefficients anisotropy. Th e approach may be also applied to magnetostatics problems as well as i n some composite materials simulation.