Overlapping additive Schwarz preconditioners for isotropic elliptic problems with degenerate coefficients

The degenerate isotropic boundary value problem –∇(ω2(x)∇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 ω2(ξ) = ξα, where α ≥ 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 α ≠ 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.