Legendre Gauss spectral collocation for the Helmholtz equation on a rectangle

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).