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Schur complement domain decomposition algorithms for spectral methods. (English) Zbl 0687.65106
Schur complement domain decomposition algorithms for spectral methods are considered. Both the Funaro-Maday-Patera weak $$C^ 1$$ matching on the interfaces [cf. A. T. Patera, J. Comput. Phys. 54, 468-488 (1984; Zbl 0535.76035)] and S. A. Orszag’s exact $$C^ 1$$ matching [ibid. 37, 70-92 (1980; Zbl 0476.65078)] are considered. Numerical results show that the condition number of the Schur complement system is of order $$O(n^ 2)$$. It is shown how this can be improved to nearly O(1) by a boundary probe preconditioned.
Reviewer: W.Heinrichs

##### MSC:
 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65Y05 Parallel numerical computation 65F35 Numerical computation of matrix norms, conditioning, scaling
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##### References:
 [1] Bjørstad, P.E.; Widlund, O.B., Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. numer. anal., 23, 1097-1120, (1986) · Zbl 0615.65113 [2] Chan, T.F., Analysis of preconditioners for domain decomposition, SIAM J. numer. anal., 24, 382-390, (1987) · Zbl 0625.65100 [3] Chan, T.F.; Goovaerts, D., Domain decomposition methods with inexact subdomain solves, () [4] Chan, T.F.; Resasco, D., A survey of preconditioners for domain decomposition, () [5] Concus, P.; Golub, G.H.; O’Leary, D.; Bunch, J.; Rose, D., A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations., Sparse matrix computation, 309-322, (1976), New York [6] Curtis, A.R.; Powell, M.J.D.; Reid, J.K., On the estimation sparse Jacobian matrices, J. inst. math. appl., 13, 117-119, (1974) · Zbl 0273.65036 [7] Dryja, M., A capacitance matrix method for Dirichlet problem on polygonal region, Numer. math., 39, 51-64, (1982) · Zbl 0478.65062 [8] Funaro, D.; Quarteroni, A.; Zanoli, P., An iterative procedure with interface relaxation for domain decomposition methods, () [9] Golub, G.H.; Mayers, D., The use of pre-conditioning over irregular regions, () · Zbl 0564.65067 [10] Gottlieb, D.; Hussaini, M.Y.; Orszag, S.A., Theory and applications of spectral methods, () [11] Orszag, S.A., Spectral methods for problems in complex geometries, J. comput phys., 37, 70-92, (1980) · Zbl 0476.65078 [12] Patera, A.T., A spectral element method for fluid dynamics: laminar flow in channel expansions, J. comput. phys., 54, 468-488, (1984) · Zbl 0535.76035 [13] Quarteroni, A.; Sacchi Landriani, G., Domain decomposition preconditioners for the spectral collocation method, () · Zbl 0675.65116 [14] Schwarz, H.A., Über eine grenzübergang durch alternirendes verfahren, (), 133-143
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