ULTRACONVERGENCE OF FINITE ELEMENT METHOD BY RICHARDSON EXTRAPOLATION FOR ELLIPTIC PROBLEMS WITH CONSTANT COEFFICIENTS

In this article, two novel Richardson extrapolation operators P-1(k) and P-2(k) 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 x(0) is an interior mesh node of the underlying mesh which is away from the boundary for a fixed distance unchanging with further mesh re finement. We show that, for both tensor product Q(k) element and simplicial P-k element, it holds vertical bar (u - P(1)(k)u(h)) (x(0))vertical bar <= ch(2k) vertical bar ln h vertical bar((k) over bar +1) and vertical bar(del u - P-2(k) ((del) over baru(h))) (x(0))vertical bar <= ch(2k) vertical bar ln h vertical bar((k) over bar +1), where u(h) is the finite element approximation of u, del is defined in section 1.1, and (k) over bar = 1 if k = 1 and (k) over tilde = 0 if k > 1. Numerical results are provided to demonstrate the theoretic findings.