Legendre Gauss Spectral Collocation for the Helmholtz Equation on a Rectangle

A spectral collocation method with collocation at the Legendre Gauss points is discussed for solving the Helmholtz equation −Δu+κ(x,y)u=f(x,y) on a rectangle with the solution u subject to inhomogeneous Robin boundary conditions. The convergence analysis of the method is given in the case of u satis...

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Bibliographic Details
Published inNumerical algorithms Vol. 36; no. 3; pp. 203 - 227
Main Authors Bialecki, Bernard, Karageorghis, Andreas
Format Journal Article
LanguageEnglish
Published New York Springer Nature B.V 01.07.2004
Subjects
Algorithms
Boundary conditions
Collocation methods
Conjugate gradient method
Dirichlet problem
Helmholtz equations
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ISSN1017-1398
1572-9265
DOI10.1023/B:NUMA.0000040056.52424.49

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Summary:A spectral collocation method with collocation at the Legendre Gauss points is discussed for solving the Helmholtz equation −Δu+κ(x,y)u=f(x,y) on a rectangle with the solution u subject to inhomogeneous Robin boundary conditions. The convergence analysis of the method is given in the case of u satisfying Dirichlet boundary conditions. A matrix decomposition algorithm is developed for the solution of the collocation problem in the case the coefficient κ(x,y) is a constant. This algorithm is then used in conjunction with the preconditioned conjugate gradient method for the solution of the spectral collocation problem with the variable coefficient κ(x,y).
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ISSN:1017-1398
1572-9265
DOI:10.1023/B:NUMA.0000040056.52424.49