A Fourth Order Hermitian Box-Scheme with Fast Solver for the Poisson Problem in a Square

A new fourth order box-scheme for the Poisson problem in a square with Dirichlet boundary conditions is introduced, extending the approach in [17]. The design is based on a “hermitian box” approach, combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2h. The goal is twofold: first to show that fourth order accuracy is obtained both for the unknown and its gradient; second, to describe a fast direct algorithm, based on the Sherman-Morrison formula and the Fast Sine Transform. Several numerical results in a square are given, indicating an asymptotic O(N log 2 (N)) computing complexity. MSC Subject Classification: 35J25 65M15 65N30 76M12 76M20