Abstract
We formulate new optimal (fourth) order one step nodal cubic spline collocation methods for the solution of various elliptic boundary value problems in the unit square. These methods are constructed so that the respective collocation equations can be solved using matrix decomposition algorithms (MDAs). MDAs are fast, direct methods which employ fast Fourier transforms and require O(N2 log N) operations on an $N \times N$ uniform partition of the unit square. The results of numerical experiments exhibit expected global optimal orders of convergence as well as desired superconvergence phenomena.
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