An efficient sixth-order solution for anisotropic Poisson equation with completed Richardson extrapolation and multiscale multigrid method

We present an efficient numerical method for anisotropic Poisson equations. The sixth-order accuracy is achieved through applying completed Richardson extrapolation on two fourth-order solutions computed from different scale grids with unequal mesh size discretization. Theoretical analysis is conducted to demonstrate that the Richardson extrapolation is able to obtain a sixth-order solution by removing the leading truncation error terms of the fourth-order solution from grid with unequal mesh sizes. The gain in efficiency is obtained through adopting partial semi-coarsening multigrid method to solve the resulting linear systems and multiscale multigrid computation to speed up the whole solution. Numerical experiments are conducted to verify the accuracy and efficiency of the proposed method and the results are compared with the existing fourth-order methods for solving 2D and 3D anisotropic Poisson equations.