Some mixed finite element methods for biharmonic equation

Some perturbed mixed finite element methods related to the reduced integration technique are considered for solving the biharmonic equation problem. On a rectangular mesh, a similar scheme was proposed in Malkus and Hughes (Comput. Methods Appl. Mech. Eng. 15 (1978) 63–81) and its convergence was analyzed in Johnson and Pitkäranta (Math. Comp. 38 (1982) 375–400). Here we modify the scheme proposed in Malkus and Hughes (1978) and prove the optimal order error estimate without the extra smoothness assumption on the solution made in Johnson and Pitkäranta (1982). On a triangular mesh, an analogous scheme is studied, and an order error estimate is proved. Some numerical results are given to show the convergence behavior of the numerical solutions.