The Multigrid Method for the Combined Hybrid Element of Linear Elasticity Problem

Combined hybrid element method is one kind of stable finite element discrete method in which the famous Babuska-Brezzi condition is satisfied automatically. So, the method is more widely used, compared with other kinds of mixed/hybrid element methods. In this paper, we develop a non-nested multigrid algorithm for combined hybrid quadrilateral or hexahedron elements of linear elasticity problem. The critical ingredient in the algorithm is a proper intergrid transfer operator. We establish such an operator on quadrilateral or hexahedron meshes and prove the mesh-independent convergence of the kth level iteration and full multigrid algorithm in L2 norm. Numerical experiments are reported to support our theoretical results and illustrate the efficiency of the developed methods. We also give the numerical experiments showing the convergence of the developed method as Poisson's ratio is close to 0.5.