Interface Schur complement preconditioning for piece wise orthotropic discretizations with high aspect ratios

Vadim Korneev

Dec. 12, 2006, 2:30 p.m. T 1010

The aim of this lecture is to present an almost optimal in the total computational work DD (domain decomposition) Dirichlet-Dirichlet type algorithm for a piece wise orthotropic discretizations on a domain, composed of rectangles with arbitrary aspect ratios. The two nonzero entries in the diagonal matrix of coefficients before products of first order derivatives in the energy integral of the problem are assumed to be arbitrary positive numbers different for each subdomain. The rectangular mesh of the finite element discretization is uniform on each subdomain of the decomposition and otherwise arbitrary.

The main problem in designing the algorithm is the interface Schur complement preconditioning, which is closely related to obtaining boundary norms for discrete harmonic functions on the shape irregular domains. The computational cost of the designed Schur complement algorithm is $O(N (log N)^{1/2}$ arithmetic operations, where N is the number of unknowns. The DD algorithm, incorporating the suggested Schur complement preconditioner, requires the same arithmetical work.