New error estimates of bi-cubic Hermite finite element methods for biharmonic equations

For the global superconvergence over the entire solution domain originated by Lin and his colleagues [7,8], this paper gives a framework of new estimates and new proofs for the basic estimates for bounds of ∫∫Ω(u−uI)xxvxx ds, ∫∫Ω(u−uI)xyvxy ds and ∫∫Ω(u−uI)xxvyy ds, which reveal more intrinsic characteristics and easier understanding and better readable. Suppose that the solution is smooth enough and the solution domain can be split into quasiuniform rectangular elements □ij with the maximal boundary length h. The study of [7,8] dealt with only the clamped boundary condition for biharmonic equations, to obtain the global superconvergence O(h4) in H2 norms under uniform rectangles □ij for the solution u∈H6(Ω). This paper is devoted to other kinds of important boundary conditions, such as the simple support condition, the natural boundary condition and their mixed types where the different boundary conditions are subject to different edges of ∂Ω. New error estimates are derived theoretically, and verified numerically to reach the global superconvergence O(h3.5) and O(h4) for different boundary conditions on different edges of Ω under uniform □ij. Note that the new superconvergence estimates in this paper are essential in practical applications, because different boundary conditions are needed in 3D blending surfaces [5] and in the combined methods for singularity problems [6].