New error estimates of Adini's elements for Poisson's equation

Abstract

In this paper, we report some new discoveries of Adini's elements for Poisson's equation in error estimates, stability analysis and global superconvergence. It is well known that the optimal convergence rate ‖ u− u h ‖ 1=O( h 3| u| 4) can be obtained, where u h 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 supercloseness ‖ u A I − u h ‖ 1=O( h 3.5‖ u‖ 5) can be obtained for uniform rectangles □ ij , where u A I 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 ( u n ) ij = g ij on the boundary are proposed to yield the Adini's solution u ∗ h having supercloseness ‖u A I−u ∗ h‖ 1= O(h 4‖u‖ 5) . Hence, the global superconvergence ‖u−Π 5u ∗ h‖ 1= O(h 4‖u‖ 5) can be achieved, where Π 5u ∗ 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. Numerical experiments in this paper are also provided to verify the supercloseness and superconvergences, O( h 3.5) and O( h 4), and the standard condition number O( h −2). It is worthy pointing out that for the Neumann problems on rectangular domains, the traditional finite element method is not as good as the newly proposed method interpolating the Neumann condition in this paper. Not only is the new method more accurate, but also economical in computation, as the discrete system has less unknowns.