An economical cascadic multigrid method for the weak Galerkin finite element approximation of second order elliptic problems

摘要

In this paper, we present both error and computational complexity estimates of economical cascadic multigrid solver for the linear system of equations arising from weak Galerkin finite element approximation of second order elliptic problems on triangular meshes. Our analysis shows that the proposed economical cascadic multigrid method with the conjugate gradient smoother is optimal in both accuracy and computational complexity for two dimensional problems. In addition, an elimination technique is utilized to further improve the computation efficiency. The interior degrees of freedom on each element are removed from the resulting linear system so that the size of the problem becomes much smaller. We demonstrate the accuracy and efficiency of our proposed economical cascadic multigrid methods through ample numerical experiments. The numerical results show that compared with the usual cascadic multigrid method, the elimination economical cascadic multigrid method saves computational cost significantly. (C) 2018 Elsevier B.V. All rights reserved.