Extrapolation multiscale multigrid method for solving 2D Poisson equation with sixth order compact scheme

Abstract

We present an extrapolation multiscale multigrid (EMMG) algorithm to solve the large linear systems arising from a sixth order compact discretization of the two dimensional Poisson equation, based on multigrid method and an extrapolation operator. With the help of Taylor expansion and interpolation theory, we develop three mid-point extrapolation formulas and combine it with the classical Richardson extrapolation strategy to design an extrapolation operator. Applying this proposed extrapolation operator for the sixth order difference solutions on the finest and finer grids, which have been computed by V-cycle multigrid method, we can construct an eighth order accurate extrapolation solution on the entire finest grid directly and efficiently. Moreover, we discuss the error of EMMG method in theoretically, and conduct some numerical experiments on square or reentrant domains, to verify that our EMMG algorithm can achieve eighth order convergence and keep less cost simultaneously.