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One-dimensional Galerkin methods and superconvergence at interior nodal points. (English) Zbl 0571.65078
The interior superconvergence is investigated for the Ritz-Galerkin method applied to m-th order self-adjoint boundary value problems and for the collocation method applied to (arbitrary) m-th order boundary value problems. This phenomenon occurs at the properly shifted zeros of the Jacobi polynomial $$P_ n^{m,m}(\sigma)$$, $$n=k+1-2m$$, k is the degree of the finite element space. The derivative of these solutions turns to be also superconvergent at the properly shifted zeros of $$P_{n+1}^{m- 1,m-1}(\sigma)$$. Both orders are one order better than the corresponding optimal orders of convergence.
Reviewer: K.Balla

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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