Variable V-cycle multigrid preconditioners for the discrete systems from combined hybrid quadrilateral elements

Combined hybrid element method is an absolutely convergent and stable hybrid element method, in which approximation spaces for field variables avoid the restriction of Babuska-Brezzi condition. So the spaces can be adopted in a larger range, compared with other kinds of mixed∕hybrid element methods. In this paper, two variable V-cycle multigrid preconditioners are proposed for the algebraic system resulting from combined hybrid quadrilateral element discretization of linear elasticity problem. Based on two kinds of simple intergrid transfer operators, constructed on quadrilateral meshes, we prove that the condition numbers of the preconditioned systems are bounded by a constant independent of the mesh size and the number of levels. A numerical example is given to indicate the asymptotically optimal performance of the algorithm.