Abstract
An algorithm for approximating certain classes of elliptic partial differential equations on a rectangle is presented. The algorithm uses high-order 9-point difference approximations to the Helmholtz-type (fourth-order) or Poisson (sixth-order) equations and the fast Fourier transform. Compared to efficient second-order fast direct, methods for smooth problems, the execution time is reduced by a large factor, typically 50 for the Helmholtz-type equations and over 100 for the Poisson problem. Comparisons with two high-order fast direct methods indicate the superiority of the algorithm.
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