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On the accuracy of the finite volume element method for diffusion equations on composite grids. (English) Zbl 0707.65073

Consider the elliptic problem $-\nabla \left(A\left(x,y\right)\nabla w\right)=f\left(x,y\right)$ in ${\Omega }={\left(0,1\right)}^{2}$, $w=0$ on $\partial {\Omega }$, which is equivalent with: Find $w\in {H}_{0}^{2}\left({\Omega }\right)$ such that, for any admissible volume $V\subset \overline{{\Omega }}$

$\left(1\right)\phantom{\rule{1.em}{0ex}}-{\int }_{\partial V}\left(A\left(x,y\right)\nabla x\right)\stackrel{\to }{n}dS={\int }_{V}fdV·$

Then, the finite volume element (FVE) method for approximating the solution (1) consists of defining a similar problem in a finite-dimensional subspace $U\subset {H}_{0}^{1}\left({\Omega }\right)$ for a finite set of volumes ${\left\{{V}_{\alpha \beta }\right\}}_{\left(\alpha ,\beta \right)}$, ($\alpha$,$\beta$)$\in S$, for a given S: Find $u\in U$ such that

$\left(2\right)\phantom{\rule{1.em}{0ex}}\forall \left(\alpha ,\beta \right)\in S,\phantom{\rule{1.em}{0ex}}-{\int }_{\partial {V}_{\alpha \beta }}\left(A\left(x,y\right)\nabla u\right)\stackrel{\to }{n}dS={\int }_{{V}_{\alpha \beta }}fdV·$

We denote $e\left(p\right)=u\left(p\right)-w\left(p\right)$, the discretization error; $p=\left({x}_{\alpha },{y}_{\beta }\right)\in G$, where u and w are the solutions of (1) and (2), respectively, and G is the composite grid, $G\subset {\Omega }$. A first evaluation error result is obtained by: If $w\in {H}_{0}^{m}\left({\Omega }\right)$ and $A\in {W}_{\infty }^{m-1}\cap {C}^{m-2}\left({\Omega }\right)$, $m=2$ or 3, then:

${\parallel e\parallel }_{1,G}{\le C\left(\left(2h\right)}^{m-1}{\parallel w\parallel }_{m,{\Omega }-{{\Omega }}_{F}}+{h}^{m/2}{\parallel w\parallel }_{m,{{\Omega }}_{F}}\right)·$

An improved error is given by: If $w\in {H}_{0}^{m}\left({\Omega }\right)$, where $m=2$ or 3, then:

${\parallel e\parallel }_{1,G}\le C{\left(2h\right)}^{m-1}{|w|}_{m,{\Omega }\setminus {{\Omega }}_{F}^{+}}+{h}^{m-1}{|w|}_{m,{{\Omega }}_{F}}\right),$

where C is a constant independent of the mesh size ${h,\parallel ·\parallel }_{1,G}$; ${\parallel ·\parallel }_{m,}$; ${|·|}_{m,}$; are the Sobolev norm and seminorm, respectively, implicitly given in the paper. The paper contains a detailed presentation of the FVE method.

Reviewer: T.Potra
##### MSC:
 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N38 Boundary element methods (BVP of PDE) 65N15 Error bounds (BVP of PDE) 35J25 Second order elliptic equations, boundary value problems