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On exact result in the finite element method. (English) Zbl 1066.65126
First, the authors consider the finite element method for a general one-dimensional boundary value problem described by a system of linear differential equations with variable coefficients and prove that no discretization error occurs at nodal points if the shape functions are considered. How to choose them it is illustrated on an example. Then they solve the analogous question for the biharmonic problem with mixed boundary conditions. Let \(u\) be the weak solution. It is shown that the Galerkin approximation of \(u\) based on biharmonic finite elements is independent of the values of \(u\) in the interior of any subelement.
Reviewer: Jan Zítko (Praha)
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
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