Abstract
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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Bialecki, B., Karageorghis, A. Legendre Gauss Spectral Collocation for the Helmholtz Equation on a Rectangle. Numerical Algorithms 36, 203–227 (2004). https://doi.org/10.1023/B:NUMA.0000040056.52424.49
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DOI: https://doi.org/10.1023/B:NUMA.0000040056.52424.49