A Multigrid Algorithm For A Robust Finite Element Method For Solving Darcy-Stokes Problems

This paper proposes a multigrid algorithm for a robust finite element method which is introduced in [20] for solving Darcy-Stokes problems.In this paper, we first introduce the variational form and the corresponding dis-crete form of the Darcy-Stokes problem, and define the corresponding norm, space, and operators. Next, we present the grid transfer operator and present our multigrid algorithm. Based on a series of lemmas and theorems, we prove the smoothness and the approximation, and obtain the convergence of multigrid algorithm. We prove that under the||·||0and||·||ι2norms the error is inverse proportional to smooth frequen-cies. Therefore, when we continue to increase times of smoothness, reasonaly, under the||·||o and||·||ι2norms the error can approach zero. Finally, numerical experiments are presented to confirm our theoretical results.