Superconvergence results on mildly structured triangulations

Abstract

Many results on superconvergence for recovered gradients of piecewise linear Galerkin approximations on triangular mesh partitions to the weak solutions of elliptic boundary value problems in two dimensions have been proved in recent years. These were obtained first for ∞-regular (fully-structured) partitions in which the mid-points of the diagonals of all quadrilaterals formed by pairs of adjacent elements are coincident. This condition was then relaxed to allow for strongly-regular meshes, in which the distance between the above mid-points is O( h 2), h being the mesh size parameter. In this paper these conditions are weakend still further to the case of globally mildly structured meshes, where the mid-point distance is O( h 1+ α ), 0< α<1, and to meshes of this type where locally α=0. After a review of recovery and gradient superconvergence, a unified approach is presented in terms of a generic gradient recovery operator which possesses specific properties on rectangular domains. Then the well-known classic theorem of Oganesyan and Rukhovets is extended to the case of mildly structured triangulations of polygonal approximations of C 3( d) domains. A class of gradient recovery operators is described on these mildly structured meshes and, using the extended Oganesyan–Rukhovets theorem, superconvergence is proved. We also obtain global superconvergence results for the recovered gradients over plane polygonal domains patchwise partitioned by fully-structured meshes. A feature of our results is that they allow local refinements of such meshes without loss of superconvergence. For the sake of completeness we have referenced the works of others in order to demonstrate the place of our work in the field.