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On the finite volume element method. (English) Zbl 0731.65093
The author considers the problem $$-\nabla \cdot (A\nabla u)=f$$ on a polygonal domain $$\Omega \subset {\mathbb{R}}^ 2$$ with $$u=0$$ on $$\Gamma_ 0$$, $$A\nabla u\cdot n=g$$ on $$\Gamma_ 1$$, $$\Gamma_ 0\cup \Gamma_ 1=\partial \Omega$$, A uniformly elliptic. The author considers piecewise linear functions v on a regular triangularization of $$\Omega$$, $$b_{ij}(v)=-\int_{\gamma_{ij}}(A\nabla v)\cdot n_{ij} ds$$ for $$\gamma_{ij}$$ an edge connecting triangle interior points (consistently taken as either circumcenters, orthocenters, incenters, or centroids), and the linear operator B defined by $$(Bv)_ i=\sum_{j}b_{ij}(v).$$ He gives conditions under which B will be uniformly elliptic, and under those conditions derives estimates on the discretization error.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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##### References:
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