One-Dimensional Galerkin Methods and Superconvergence at Interior Nodal Points

摘要

A minor kind of superconvergence at specific points inside the segments of the partition is discussed for two classes of Galerkin methods: the Ritz-Galerkin method for 2m-th order self-adjoint boundary problems and the collection method for arbitrary m-th order boundary problems. These interior points are the zeros of the Jacobi polynomial P(m,m)sub n (sigma) shifted to the segments of the partition; n = k+1 - 2m, where k is the degree of the finite element space. The order of convergence at these points is k+2, one order better than the optimal order of convergence. Also, it can be proved that the derivative of the finite element solution is superconvergent of O(h(k+1)) at the zeros of the Jacobi polynomial P(m-1,m-1) sub n+1(sigma) shifted to the segments of the partition. This is one order better than the optimal order of convergence for the derivative.