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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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