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The degenerate isotropic boundary value problem –∇(ω 2 ( 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 ω 2 (ξ) = ξ α , 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.

Overlapping additive Schwarz preconditioners for isotropic elliptic problems with degenerate coefficients