\(\eta\%\)-superconvergence in the interior of locally refined meshes of quadrilaterals: Superconvergence of the gradient in finite element solutions of Laplace's and Poisson's equations (Q1344317)

The paper deals with the concept of \(\eta\%\)-superconvergence of finite element solutions. A point of \(\eta\%\)-superconvergence is a point where the approximation error, in the limit when refining the mesh, does not exceed \(\eta\%\) of the maximal approximation error in the respective element. Classical superconvergence corresponds to the case \(\eta= 0\). A computer based approach of determining regions of \(\eta\%\)-convergence for the gradient in boundary problems for the Poisson equation is presented. Sensitivity to mesh topology is discussed. Several numerical studies are presented. The paper is the third in a series of three on superconvergence.