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On the finite volume element method. (English) Zbl 0731.65093
The author considers the problem $-\nabla ·\left(A\nabla u\right)=f$ on a polygonal domain ${\Omega }\subset {ℝ}^{2}$ with $u=0$ on ${{\Gamma }}_{0}$, $A\nabla u·n=g$ 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 ${\Omega }$, ${b}_{ij}\left(v\right)=-{\int }_{{\gamma }_{ij}}\left(A\nabla v\right)·{n}_{ij}ds$ 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 ${\left(Bv\right)}_{i}={\sum }_{j}{b}_{ij}\left(v\right)·$ He gives conditions under which B will be uniformly elliptic, and under those conditions derives estimates on the discretization error.

##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N15 Error bounds (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems
##### References:
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