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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.

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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