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Triangular elements in the finite element method

Authors: James H. Bramble and Miloš Zlámal
Journal: Math. Comp. 24 (1970), 809-820
MSC: Primary 65.66
MathSciNet review: 0282540
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Abstract: For a plane polygonal domain $\Omega$ and a corresponding (general) triangulation we define classes of functions ${p_m}(x,y)$ which are polynomials on each triangle and which are in ${C^{(m)}}(\Omega )$ and also belong to the Sobolev space $W_2^{(m + 1)}(\Omega )$. Approximation theoretic properties are proved concerning these functions. These results are then applied to the approximate solution of arbitrary-order elliptic boundary value problems by the Galerkin method. Estimates for the error are given. The case of second-order problems is discussed in conjunction with special choices of approximating polynomials.

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Keywords: Finite element method, Ritz method, Galerkin method, piecewise polynomial subspaces, approximation of solution, elliptic boundary problems
Article copyright: © Copyright 1970 American Mathematical Society