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Quasi-optimal Schwarz methods for the conforming spectral element discretization. (English) Zbl 0889.65123
The spectral element method is used for three-dimensional selfdajoint elliptic equations. The domain is decomposed into hexahedral spectral elements. A conforming Galerkin method is used where the integrals are approximated by Gauss-Lobatto-Legendre quadrature. The resulting linear systems are solved by a preconditioned conjugate gradient method. Techniques for nonregular meshes are also developed. Two finite element preconditioners are discussed: the wirebasket method of B. F. Smith [Numer. Math. 60, No. 2, 219-234 (1991; Zbl 0724.65110)] and the overlapping Schwarz algorithm. Numerical experiments demonstrate the efficiency of these methods.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
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