In this paper, we consider the degenerated isotropic boundary value problem −∇(ω 2 (x)∇u(x, y)) =... more In this paper, we consider the degenerated isotropic boundary value problem −∇(ω 2 (x)∇u(x, y)) = f (x, y) on the unit square (0, 1) 2 . 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-angled triangles. The system of linear algebraic equations is solved by a preconditioned 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 preoconditioning operation requires O(N ) operations, where N is the number of unknowns. Several numerical experiments show the preformance of the proposed method.
The problem of interpolation of scattered data on the unit sphere has many applications in geodes... more The problem of interpolation of scattered data on the unit sphere has many applications in geodesy and Earth science in which the sphere is taken as a model for the Earth. Spherical radial basis functions provide a convenient tool for constructing the interpolant. However, the underlying linear systems tend to be ill-conditioned. In this paper, we present an additive Schwarz preconditioner for accelerating the solution process. An estimate for the condition number of the preconditioned system will be discussed. Numerical experiments using MAGSAT satellite data will be presented.
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