New error estimates of Adiniʹs elements for Poissonʹs equation Original Research Article

In this paper, we report some new discoveries of Adiniʹs elements for Poissonʹs equation in error estimates, stability analysis and global superconvergence. It is well known that the optimal convergence rate ‖u−uh‖1=O(h3|u|4) can be obtained, where uh and u are the Adiniʹs solution and the true solution, respectively. In this paper, for all kinds of boundary conditions of Poissonʹs equations, the supercloseness ‖uAI−uh‖1=O(h3.5‖u‖5) can be obtained for uniform rectangles □ij, where uAI is the Adiniʹs interpolation of the true solution u. Moreover, for the Neumann problems of Poissonʹs equation, new treatments adding the explicit natural constraints (un)ij=gij on the boundary are proposed to yield the Adiniʹs solution View the MathML source having supercloseness View the MathML source. Hence, the global superconvergence View the MathML source can be achieved, where View the MathML source is an a posteriori interpolant of polynomials with order five based on the obtained solution View the MathML source. New basic estimates of errors are derived for Adiniʹs elements. Numerical experiments in this paper are also provided to verify the supercloseness and superconvergences, O(h3.5) and O(h4), 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.