A spectral approximation for the Poisson equation defined on Ω = ]−1, 1[×] −1,1[ is studied. The domain Ω 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 in order to obtain a unique solution from the resulting linear system. Different values of the discretization parameters are allowed in each rectangle. We prove the stability of the scheme and give convergence estimates. The rate of convergence in a single subdomain, depends only on the regularity of the exact solution therein. An efficient preconditioning matrix is proposed.
A multidomain spectral approximation of elliptic equations / Funaro, D.. - In: NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0749-159X. - 2:3(1986), pp. 187-205. [10.1002/num.1690020304]
A multidomain spectral approximation of elliptic equations
Funaro D.
1986
Abstract
A spectral approximation for the Poisson equation defined on Ω = ]−1, 1[×] −1,1[ is studied. The domain Ω 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 in order to obtain a unique solution from the resulting linear system. Different values of the discretization parameters are allowed in each rectangle. We prove the stability of the scheme and give convergence estimates. The rate of convergence in a single subdomain, depends only on the regularity of the exact solution therein. An efficient preconditioning matrix is proposed.
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