Overlapping additive Schwarz preconditioners for isotropic elliptic problems with degenerate coefficients

The degenerate isotropic boundary value problem -del(omega(2)(x)del u(x,y)) = f(x,y) on the unit square (0, 1)(2) is considered in this paper. The weight function is assumed to be of the form omega(2)(xi) = xi(alpha), where alpha >= 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right triangles. The system of linear algebraic equations is solved by a preconditioned conjugate gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for alpha not equal 1. The preconditioning operation requires O(N) operations, where N is the number of unknowns. Several numerical experiments show the performance of the proposed method.