An efficient extrapolation full multigrid method for elliptic problems in two and three dimensions

Abstract

An extrapolation full multigrid (EXFMG) method is proposed for solving the large linear systems arising from linear finite element discretization of two- (2D) and three-dimensional (3D) elliptic boundary value problems. A good initial guess on the next finer grid is constructed through combining Richardson extrapolation and quadratic FE interpolation for the numerical solutions on two-level of grids (current and previous grids). And the FE linear system on the next finer grid is solved to obtain the EXFMG solution with discretization-level accuracy by performing one or two multigrid cycles with Gauss–Seidel smoother. Moreover, Richardson extrapolation is further used to achieve higher-order accuracy. The numerical results of five typical elliptic problems including one 2D problem and four 3D problems are conducted to show the effectiveness and efficiency of the EXFMG algorithm.