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Additive Schwarz domain decomposition with a radial basis approximation. (English) Zbl 1051.65121
Summary: Based on the idea of additive Schwarz domain decomposition, this paper gives the theoretical justification to combine the compactly supported radial basis functions with those advanced numerical techniques in space decomposition, iterative subspace correction, and preconditioning. This overcomes the ill-conditioning problem resulted from using the radial basis function as a global interpolant.
The preconditioner for the self-adjoint elliptic operator is constructed and an analogous order estimate on the preconditioner similar to the finite element method is given. Since the radial basis functions are smooth and spatial dimension independent, its extension to solve high order differential equations and high dimensional problems has been proven to be achievable.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
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