Global superconvergence of finite element methods for biharmonic equations and blending surfaces

Abstract

In this paper, new numerical algorithms of finite element methods (FEM) are reported for both biharmonic equations and 3D blending surfaces, to achieve the global superconvergence O( h 3)- O( h 4) in H 2 norms. This is significant, compared with the optimal convergence O( h 2). The algorithms are simple because only an a posteriori interpolant solution is needed. Such a global convergence method was originated by Lin and his colleagues in [1–3] for only the clamped boundary conditions. Recently, we extended the global superconvergence to other boundary conditions, such as the simple support condition, the periodic boundary, and the natural boundary condition. Moreover, we apply in [4–6] this global superconvergence to the FEM using the penalty techniques for biharmonic equations and blending problems, also to reach O( h 3) and O( h 4) for quasiuniform and uniform □ ij , respectively. Currently, we develop in [7,8] and in this paper the FEM using the penalty plus hybrid techniques to reduce the condition number down to O( h −4)- O( h −5) of the associated matrix, while retaining superconvergence O( h 3)- O( h 4). Since instability is severe for biharmonic equations, any reduction of the condition number is crucial. By the new algorithms in this paper, not only can a great deal of CPU time be saved, but also the complicated biharmonic equations and blending surfaces may be solved in double precision. Numerical experiments are carried out to support the theoretical conclusions.