Matrix decomposition algorithms for modified spline collocation for Helmholtz problems
Date
2003ISSN
1064-8275Source
SIAM Journal on Scientific ComputingVolume
24Issue
5Pages
1733-1753Google Scholar check
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We consider the solution of various boundary value problems for the Helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal (fourth-) order approximations. For the solution of the collocation equations, we formulate matrix decomposition algorithms, fast direct methods which employ fast Fourier transforms and require O(N2 log N) operations on an N × N uniform partition of the unit square. A computational study confirms the published analysis for the Dirichlet problem and indicates that similar results hold for Neumann, mixed, and periodic boundary conditions. The numerical results also exhibit superconvergence phenomena not reported in earlier studies.