A multidomain spectral approximation of elliptic equations.

*(English)*Zbl 0622.65104A spectral approximation for the Poisson equation in one and two dimensions (on a square) is studied. The domain is decomposed into two rectangular regions and the equation is collocated at the Legendre nodes in each domain. On the common boundary of the two subdomains, suitable conditions are imposed so that a unique solution is obtained for the resulting linear system. Different values of the discretization parameters are allowed in each rectangle. The stability of the scheme is established and convergence estimates given. The rate of convergence in a single subdomain depends only on the regularity of the exact solution therein. An efficient preconditioning matrix is proposed. Suggestions for further research are added.

Reviewer: W.Ames

##### 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 |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

##### Keywords:

domain decomposition; collocation; spectral approximation; Poisson equation; Legendre nodes; stability; rate of convergence; preconditioning
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\textit{D. Funaro}, Numer. Methods Partial Differ. Equations 2, 187--205 (1986; Zbl 0622.65104)

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