The local ultraconvergence for high-degree Galerkin finite element methods

In this paper, we investigate the local ultraconvergence of k  -degree (k≥3)(k≥3) finite element methods for the second order elliptic boundary value problem with constant coefficients over a family of uniform rectangular/triangular meshes ThTh on a bounded rectangular domain D. The k  -degree finite element estimates are developed for the Green's function and its derivatives. They are employed to explore the relationship among dist(x,∂D), dist(x,M)dist(x,∂D), dist(x,M) and the ultraconvergence of k-degree finite element methods at vertex x, where M is the set of corners of D. Numerical examples are conducted to demonstrate our theoretical results.