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Zbl 0731.65093
Cai, Zhiqiang
On the finite volume element method.
(English)
[J] Numer. Math. 58, No.7, 713-735 (1991). ISSN 0029-599X; ISSN 0945-3245/e

The author considers the problem $-\nabla \cdot (A\nabla u)=f$ on a polygonal domain $\Omega \subset {\bbfR}\sp 2$ with $u=0$ on $\Gamma\sb 0$, $A\nabla u\cdot n=g$ on $\Gamma\sb 1$, $\Gamma\sb 0\cup \Gamma\sb 1=\partial \Omega$, A uniformly elliptic. The author considers piecewise linear functions v on a regular triangularization of $\Omega$, $b\sb{ij}(v)=-\int\sb{\gamma\sb{ij}}(A\nabla v)\cdot n\sb{ij} ds$ for $\gamma\sb{ij}$ an edge connecting triangle interior points (consistently taken as either circumcenters, orthocenters, incenters, or centroids), and the linear operator B defined by $(Bv)\sb i=\sum\sb{j}b\sb{ij}(v).$ He gives conditions under which B will be uniformly elliptic, and under those conditions derives estimates on the discretization error.
[J.R.Kuttler (Laurel)]
MSC 2000:
*65N30 Finite numerical methods (BVP of PDE)
65N15 Error bounds (BVP of PDE)
35J25 Second order elliptic equations, boundary value problems

Keywords: finite volume method; finite elements; error estimates; finite volume element method

Cited in: Zbl 0913.65097

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