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Zbl 0707.65073
Cai, Zhiqiang; McCormick, Steve
On the accuracy of the finite volume element method for diffusion equations on composite grids.
(English)
[J] SIAM J. Numer. Anal. 27, No.3, 636-655 (1990). ISSN 0036-1429; ISSN 1095-7170/e

Consider the elliptic problem $-\nabla (A(x,y)\nabla w)=f(x,y)$ in $\Omega =(0,1)\sp 2$, $w=0$ on $\partial \Omega$, which is equivalent with: Find $w\in H\sp 2\sb 0(\Omega)$ such that, for any admissible volume $V\subset {\bar \Omega}$ $$(1)\quad -\int\sb{\partial V}(A(x,y)\nabla x)\vec n dS=\int\sb{V}f dV.$$ Then, the finite volume element (FVE) method for approximating the solution (1) consists of defining a similar problem in a finite-dimensional subspace $U\subset H\sp 1\sb 0(\Omega)$ for a finite set of volumes $\{V\sb{\alpha \beta}\}\sb{(\alpha,\beta)}$, ($\alpha$,$\beta$)$\in S$, for a given S: Find $u\in U$ such that $$(2)\quad \forall (\alpha,\beta)\in S,\quad - \int\sb{\partial V\sb{\alpha \beta}}(A(x,y)\nabla u)\vec n dS=\int\sb{V\sb{\alpha \beta}}f dV.$$ We denote $e(p)=u(p)-w(p)$, the discretization error; $p=(x\sb{\alpha},y\sb{\beta})\in G$, 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 $w\in H\sp m\sb 0(\Omega)$ and $A\in W\sb{\infty}\sp{m-1}\cap C\sp{m- 2}(\Omega)$, $m=2$ or 3, then: $$\Vert e\Vert\sb{1,G}\le C((2h)\sp{m- 1}\Vert w\Vert\sb{m,\Omega -\Omega\sb F}+h\sp{m/2}\Vert w\Vert\sb{m,\Omega\sb F}).$$ An improved error is given by: If $w\in H\sp m\sb 0(\Omega)$, where $m=2$ or 3, then: $$\Vert e\Vert\sb{1,G}\le C(2h)\sp{m-1}\vert w\vert\sb{m,\Omega \setminus \Omega\sb F\sp+}+h\sp{m- 1}\vert w\vert\sb{m,\Omega\sb F}),$$ where C is a constant independent of the mesh size $h,\Vert \cdot \Vert\sb{1,G}$; $\Vert \cdot \Vert\sb{m,}$; $\vert \cdot \vert\sb{m,}$; are the Sobolev norm and seminorm, respectively, implicitly given in the paper. The paper contains a detailed presentation of the FVE method.
[T.Potra]
MSC 2000:
*65N30 Finite numerical 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

Keywords: finite volume element method; diffusion equations; composite grids; discretization error

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