A fourth‐order Hermitian box‐scheme with fast solver for the Poisson problem in a cube

Abstract

We present a fourth-order Hermitian box-scheme (HB-scheme) for the Poisson problem in a cube. A singlenonstaggered regular grid is used supporting the discrete unknowns u and ∇u. The scheme is fourth-order accurate for u and ∇u in L ∞ norm. The fast numerical resolution uses a matrix capacitance method, resulting in a computational complexity of O(N 3 log 2 (N)). Numerical results are reported on several examples including nonseparable problems. The present scheme is the extension to the three-dimensional case of the HB-scheme presented in Abbas and Croisille [J Sci Comp 49 (2011), 239-267].

Analysis of a Fourth Order Scheme and Application of Local Defect Correction Method

“…This scheme appears to be fourth order accurate in practice. It has been successfully extended to two dimensions in [9] and three dimensions in [10]. Let's recall the principle of this scheme.…”

“…This gain is expected to be more interesting in higher dimensions. A generalization of this application to higher dimensions using cubic spline interpolations, based on the previous results in [9,10] will be a part of future work.…”

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

“…The scheme does not use any kind of staggered grid. All the unknowns are located at points (x i ,y j ), [23,29]. The iterative method is based on the repeated solution by a fast direct method of a discrete Poisson equation on a rectangular mesh.…”