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A finite element method scheme for one-dimensional elliptic equations with high superconvergence at the nodes. (English) Zbl 0548.65067
We set a $$P_ 1$$-type finite element method scheme to approximate one- dimensional elliptic equations. We prove that an appropriate choice of numerical integration formula improves the classical error estimates by a superconvergence result at the nodes of the mesh: $$O(h^ 4)$$ instead of $$O(h^ 2)$$. This result is actually very cheap: indeed, the integration formulas are easy (Simpson) and, moreover, the structure and the size of the linear system are the same as those of the classical scheme.

##### 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 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 34B05 Linear boundary value problems for ordinary differential equations 35J25 Boundary value problems for second-order elliptic equations
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##### References:
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