Ultraconvergence of high order FEMs for elliptic problems with variable coefficients (Q527823)

The authors consider the local ultraconvergence properties of the high-order finite element method (FEM) for second-order elliptic problems with variable coefficients. One of the main tools used in this study is the introduction of a novel local interpolation operator to post-process finite element solutions for variable-coefficient problems. Unlike the classical interpolation operator defined for instance in [\textit{Q. Lin} and \textit{J. Zhou}, Comput. Methods Appl. Mech. Eng. 196, No. 37--40, 3779--3784 (2007; Zbl 1173.65370)], the local operator interpolates the value of the original finite element solution at all vertices of the underlying mesh in a patch instead of interpolating all nodes in a relatively smaller-sized patch. Under suitable regularity and mesh conditions, it is shown that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the post-precessed \(k\)th (with \(k\geq2\)) order finite element solution converges to the gradient of the exact solution with order \(h^{k+2} (\ln h)^ 3\). The proof of this ultraconvergence property uses some new estimates for the discrete Green's function, a symmetry theory derived in [\textit{L. B. Wahlbin}, Superconvergence in Galerkin finite element methods. Berlin: Springer Verlag (1995; Zbl 0826.65092)], and the Richardson extrapolation technique. Some numerical experiments are presented to support the theoretical results.