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.

Keywords:

• Richardson extrapolation
• full multigrid algorithm
• quadratic interpolation
• finite element method
• elliptic equation

2010 AMS Subject Classifications:

• 65N22
• 65N55

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by Science Challenge Project [grant number TZ2016002], the National Natural Science Foundation of China [grant number 41874086], the Excellent Youth Foundation of Hunan Province of China [grant number 2018JJ1042], and the Natural Science Foundation of Yunnan Province of China [grant number 2017FH001-012].