Local and parallel algorithms for fourth order problems discretized by the Morley-Wang-Xu element method

This paper systematically studies numerical solution of fourth order problems in any dimensions by use of the Morley-Wang-Xu (MWX) element discretization combined with two-grid methods (Xu and Zhou (Math Comp 69:881-909, 1999)). Since the coarse and fine finite element spaces are nonnested, two intergrid transfer operators are first constructed in any dimensions technically, based on which two classes of local and parallel algorithms are then devised for solving such problems. Following some ideas in (Xu and Zhou (Math Comp 69:881-909, 1999)), the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems (Huang et al. (Appl Numer Math 37:519-533, 2001); Huang et al. (Sci China Ser A 49:109-120, 2006)), and the error estimates for the intergrid transfer operators, we prove that the discrete energy errors of the two classes of methods are of the sizes O(h + H (2)) and O(h + H (2)(H/h)((d-1)/2)), respectively. Here, H and h denote respectively the mesh sizes of the coarse and fine finite element triangulations, and d indicates the space dimension of the solution region. Numerical results are performed to support the theory obtained and to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators.