Ultraconvergence of finite element method by Richardson extrapolation for elliptic problems with constant coefficients

In this article, two novel Richardson extrapolation operators Pk1 and Pk2 are proposed to investigate local 2k order ultraconvergence properties of the kth order Lagrange finite element method for the second order elliptic problem with constant coefficients. Assume that x0 is an interior mesh node of the underlying mesh which is away from the boundary for a fixed distance unchanging with further mesh refinement. We show that, for both tensor product element and simplicial ℙk element, it holds |(u - Pk1uh)(x0)| ≤ ch |lnh| k+1 and |(u - Pk2 (u))(x0)| \u3c ch | lnh|k+1, where u the finite element approximation of u, | is defined in section 1.1, and k = 1 if k = 1 and k = 0 if k \u3e 1. Numerical results are provided to demonstrate the theoretic findings