Co-author Ningnig Yan, Institute of System Science.

In this paper, we report some new discoveries of Adini’s elements for the Neumann problems of Poisson’s equation in error estimates, stability analysis and global superconvergence. It is well known that the optimal convergence rate ku−uhk1,S = O(h3)|u|4,S can be obtained, where uh and u are the Adini’s solution and the true solution respectively. In this paper, for all kinds of boundary conditions of Poisson’s equations, the superclose ku AI − uhk1,S = O(h3.5)|u|5,S can be obtained for uniform rectangles ij , where u AI is the Adini’s interpolation of the true solution u. Moreover, for the Neumann problems of Poisson’s equation, new treatments adding the explicit natural constraints (un)ij = gij on ∂S are proposed to yield the Adini’s solution u∗h having superclose kuAI −u∗hk1,S = O(h4)|u|5,S. Hence, the global superconvergence ku − Π5pu∗hk1,S = O(h4)|u|5,S can be achieved, where Π5pu∗h is an a posteriori interpolant of polynomials with order five based on the obtained solution u∗h. New basic estimates of errors are derived for Adini’s elements, and new proofs are provided to derive the optimal rate O(h−2) of condition number for the associated matrix resulting from Adini’s elements. Numerical experiments in this paper are alsoprovided to verify perfectly the superconvergences, O(h3.5) and O(h4), and the optimal condition number O(h−2).