A C^1 Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence
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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.
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