We develop a sixth order finite difference discretization strategy to solve the two dimensional Poisson equation, which is based on the fourth order compact discretization, multigrid method, Richardson extrapolation technique, and an operator based interpolation scheme. We use multigrid V-Cycle procedure to build our multiscale multigrid algorithm, which is similar to the full multigrid method (FMG). The multigrid computation yields the fourth order accurate solution on both the fine grid and the coarse grid. A sixth order accurate coarse grid solution is computed by using the Richardson extrapolation technique. Then we apply our operator based interpolation scheme to get the sixth order accurate solution on the fine grid. Numerical experiments are conducted to show the accuracy and the efficiency of our new method, compared to Sun-Zhang's sixth order Richardson extrapolation compact (REC) discretization strategy using Alternating Direction Implicit (ADI) method and the standard fourth order compact difference (FOC) scheme using a multigrid method.
Mathematics Subject Classification:
Technical Report CMIDA-HiPSCCS 003-08, Department of Computer Science, University of Kentucky, Lexington, KY, 2008.
The research work of Jun Zhang was supported in part by NSF under grants CCF-0727600 and CCF-0527967, in part by the Kentucky Science and Engineering Foundation under grant KSEF-148-502-06-186, in part by the Alzheimer's Association under grant NIGR-06-25460, and in part by NIH under grant 1 R01 HL086644-01.