Preconditioners for pseudodifferential equations on the sphere with radial basis functions

Abstract

In a previous paper a preconditioning strategy based on overlapping domain decomposition was applied to the Galerkin approximation of elliptic partial differential equations on the sphere. In this paper the methods are extended to more general pseudodifferential equations on the sphere, using as before spherical radial basis functions for the approximation space, and again preconditioning the illconditioned linear systems of the Galerkin approximation by the additive Schwarz method. Numerical results are presented for the case of hypersingular and weakly singular integral operators on the sphere S 2 .

Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions

“…Hence K is a pseudodifferential operator of order 1. This allows us to extend our analysis [17] for the overlapping Schwarz preconditioner on the sphere to the prolate spheroid. The preconditioner is defined by the additive Schwarz operator, using a subspace decomposition of V τ as V τ = V 0 + .…”

Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

“…Thus, we can viewû h (z j ) as a Galerkin approximation toû(z j ). Concretely, to computeû h (z) = K p=1Û p (z)Φ p we form the K × K matrices B and S, with entries 15) form the load vector G(z) ∈ C K with components G p (z) = g h (z), Φ p , and then solve the K × K complex linear system 16) to obtain the solution vectorÛ(z) ∈ C K with componentsÛ p (z). In contrast to finite element mass and stiffness matrices, B and S are not sparse because the SRBFs have large supports.…”

A domain decomposition method for solving the hypersingular integral equation on the sphere with spherical splines

“…Spherical radial basis functions seem to be a better choice [18]. However, the resulting matrix system from this approximation is very ill-conditioned.…”

“…However, the resulting matrix system from this approximation is very ill-conditioned. Even though overlapping additive Schwarz preconditioners can be designed for this problem, the condition number of the preconditioned system still depends on the number of subdomains and the angles between subspaces; see [18].…”