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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 \(-\nabla (A(x,y)\nabla w)=f(x,y)\) in \(\Omega =(0,1)^ 2\), \(w=0\) on \(\partial \Omega\), which is equivalent with: Find \(w\in H^ 2_ 0(\Omega)\) such that, for any admissible volume \(V\subset {\bar \Omega}\) \[ (1)\quad -\int_{\partial V}(A(x,y)\nabla x)\vec n dS=\int_{V}f dV. \] Then, the finite volume element (FVE) method for approximating the solution (1) consists of defining a similar problem in a finite-dimensional subspace \(U\subset H^ 1_ 0(\Omega)\) for a finite set of volumes \(\{V_{\alpha \beta}\}_{(\alpha,\beta)}\), (\(\alpha\),\(\beta\))\(\in S\), for a given S: Find \(u\in U\) such that \[ (2)\quad \forall (\alpha,\beta)\in S,\quad - \int_{\partial V_{\alpha \beta}}(A(x,y)\nabla u)\vec n dS=\int_{V_{\alpha \beta}}f dV. \] We denote \(e(p)=u(p)-w(p)\), the discretization error; \(p=(x_{\alpha},y_{\beta})\in G\), 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 \(w\in H^ m_ 0(\Omega)\) and \(A\in W_{\infty}^{m-1}\cap C^{m- 2}(\Omega)\), \(m=2\) or 3, then: \[ \| e\|_{1,G}\leq C((2h)^{m- 1}\| w\|_{m,\Omega -\Omega_ F}+h^{m/2}\| w\|_{m,\Omega_ F}). \] An improved error is given by: If \(w\in H^ m_ 0(\Omega)\), where \(m=2\) or 3, then: \[ \| e\|_{1,G}\leq C(2h)^{m-1}| w|_{m,\Omega \setminus \Omega_ F^+}+h^{m- 1}| w|_{m,\Omega_ F}), \] where C is a constant independent of the mesh size \(h,\| \cdot \|_{1,G}\); \(\| \cdot \|_{m,}\); \(| \cdot |_{m,}\); 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 element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element 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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