A C1 PETROV-GALERKIN METHOD AND GAUSS COLLOCATION METHOD FOR 1D GENERAL ELLIPTIC PROBLEMS AND SUPERCONVERGENCE

In this paper, we present and study C-1 Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree k (>= 3) for one-dimensional elliptic equations. We prove that, the solution and its derivative approximations converge with rate 2k - 2 at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree k + 1 in each element, the first-order derivative approximation is superconvergent at all interior k - 2 Lobatto points, and the second-order derivative approximation is superconvergent at k - 1 Gauss points, with an order of k + 2, k + 1, and k, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in H-2, H-1, and L-2 norms. All theoretical findings are confirmed by numerical experiments.