New error estimates of Adini's elements for Poisson's equation (Q596554)

The authors investigate global superconvergence of Adini's elements for second order elliptic problems, in particular for the Poisson equation. Based on careful error estimates the authors obtain new supercloseness: for all kind of boundary conditions of Poisson's equations, the supercloseness \(| | u_{A}^{I}-u_{h}| | _{1,S}=O(h^{3.5}| | u| | _{5,S})\) can be obtained for uniform rectangles, where \(S\) is the domain of the problem and \(u_{A}^{I}\) is the Adini interpolation of the true solution \(u\); for Neumann problems of the Poisson equation, the explicit natural constraints \((u_{n})_{ij}=g_{ij}\) on \(S\) are proposed to obtain the supercloseness \(\|u_{A}^{I}-u_{h}^*\|_{1,S}=O(h^4\|u\|_{5,S})\). New error estimates with the different proofs are provided.