Modified nodal cubic spline collocation for Poisson's equation

We present a new modified nodal cubic spline collocation scheme for solving the Dirichlet problem for Poisson's equation on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the solution can be computed on an (N + 1) x (N + 1) uniform partition of the square with cost O(N(2)logN) using a direct fast Fourier transform method. Using two comparison functions, we derive an optimal fourth order error bound in the continuous maximum norm. We compare our scheme with other modified nodal cubic spline collocation schemes; in particular, the one proposed by Houstis, Vavalis, and Rice in [SIAM J. Numer. Anal., 25 (1988), pp. 54-74]. We believe that our paper gives the first correct convergence analysis of a modified nodal cubic spline collocation for solving partial differential equations.