The superconvergence gradient recovery method for linear finite element method with polygons

To identify the superconvergent areas of gradient solutions is a worthwhile part of gradient recovery in finite element method. Compared with triangular and quadrilateral elements, the position determination for the superconvergent gradient of polygonal element is much more difficult because of the non-polynomial form of interpolation functions. In this article, the gradient superconvergence points in C-0 polygonal element is discovered, and its existence is proved theoretically. By dividing the canonical polygonal element into sub-triangles, a set of provisional triangular elements is generated to replace the existing polygon element. The superconvergent results of gradient solutions are obtained at the centroids of sub-triangles. Based on that, the gradient recovery method using superconvergent patch recovery (SPR) is developed for general polygonal mesh. Numerical examples for the Poisson equation are solved and the simulated results compare with polynomial preserving recovery (PPR) to demonstrate the accuracy and convergence properties of the proposed method. The extend application of the superconvergence recovery method for adaptive analysis is also presented at the end.