R. Eymard, T. Gallouet and R. Herbin, Convergence of finite Volume schemes for semilinear convection diffusion equations, Numer. Math. 82:91-116,1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in part 2. 4 Discrete Sobolev Inequalities and L p Error Estimates. 4.1 Convergence of the finite volume scheme on admissible meshes. The existence and uniqueness of the solution (u K ) K2T to the scheme (6) 9) is an easy consequence of the following maximum principle (see [18] 12] or [10] for the proof) Proposition 1 (Maximum Principle) Under assumptions 1, let T be an admissible mesh in the sense of definition 1; and (f K ) K2T , v K;oe ) oe2EK ; K2T and (u oe ) oe2E ext be defined by (5) and (9) If fK 0 for all K 2 T , and u oe 0, for all oe 2 E ext , then the solution (u ....

Eymard R., T. Gallou et and R. Herbin , Convergence of finite volume schemes for semilinear convection diffusion equations, accepted for publication in Numer. Math.

.... meshes were obtained for linear convection diffusion equations for Dirichlet boundary conditions (see [27] 28] 21] 30] 22] 9] and Neumann or Fourier boundary conditions [9] and a convergence result (without regularity assumption) for semilinear convection diffusion equations [10]. In the following section, we introduce the meshes and some discrete functional spaces, norms and tools for these spaces which we use in our convergence proofs. In particular, we prove in Lemma 2.2 a lower bound for the lower limit of the discrete H 1 norm (see Definition 2.4) of piecewise ....

....finite volume solution to the exact one. This proof uses a compactness result which is obtained thanks to 3 an estimate on the space translates of the approximate solutions. This compactness result is adapted from one which was obtained for linear or semi linear convection diffusion equations [10], 9] The two main original points which have to be introduced here for the proof of convergence are 1. the use of a lower bound for the lower limit of the discrete H 1 norm obtained in Lemma 2.2 in order to obtain the term Gamma R Omega ru(x)ru(x)dx of the variational inequalities (5) ....

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R. Eymard, T. Gallouet and R. Herbin, Convergence of finite Volume schemes for semilinear convection diffusion equations, Numer. Math. 82:91-116,1999.

....0 is assumed. 2 Admissible (or restricted admissible) meshes include, for instance, meshes made with triangles and rectangles in two space dimensions, and also Voronoi meshes: the latter consist in building a mesh using the orthogonal bisectors from a given family of points (for more details see [7]) Admissible meshes will be used for the Neumann boundary conditions. Property (v) of the restricted admissible meshes is needed for the Dirichlet and Robin boundary conditions. 3 Dirichlet boundary conditions The first type of boundary condition which we consider is a Dirichlet condition: u(x) ....

....g D 2 L 2 ( Omega Gamma , g D is no longer defined pointwise, but (9) may be replaced by u oe = 1 m(oe) R oe g D (y) dfl(y) where dfl stands for the (d Gamma 1) dimensional Lebesgue measure. In this latter case we do not obtain an error estimate but only a convergence result as in [7]. For all K 2 T , let fK , respectively b K , denote the mean value of f , respectively of b that is to say f K = 1 m(K) Z K f(x) dx and b K = 1 m(K) Z K b(x) dx: 10) Then the considered finite volume scheme is defined by the following equations: X oe2EK i F K;oe v K;oe u oe; j ....

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R. Eymard, T. Gallou et and R. Herbin , Convergence of finite volume schemes for semilinear convection diffusion equations, accepted for publication in Numer. Math. (1999).

....E 0 is assumed. Admissible (or restricted admissible) meshes include, for instance, meshes made with triangles and rectangles in two space dimensions, and also Voronoi meshes: the latter consists in building a mesh using the orthogonal bisectors from a given family of points (for more details see [7]) Admissible meshes will be used for the Neumann boundary conditions. Property (v) of the restricted admissible meshes is needed for the Dirichlet and Robin boundary conditions. Remark 1 In the case of the operator div(kr: which is considered in Equation (2) where k is a function from Omega to ....

....2 L 2 ( Omega Gamma , g D is no longer defined pointwise, but (10) may be replaced by u oe = 1 m(oe) R oe g D (y) dfl(y) where dfl stands for the (d Gamma 1) dimensional Lebesgue measure on oe. In this latter case we do not obtain an error estimate but only a convergence result as in [7]. For all K 2 T , let fK and b K denote the mean value of f and b on K that is to say f K = 1 m(K) Z K f(x) dx and b K = 1 m(K) Z K b(x) dx: 11) Then the considered finite volume scheme is defined by the following equations: X oe2EK i F K;oe v K;oe u oe; j m(K) b K uK = m(K) ....

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R. Eymard, T. Gallou et and R. Herbin , Convergence of finite volume schemes for semilinear convection diffusion equations, accepted for publication in Numer. Math. (1999).

.... the pure diffusion case, the cell centered finite volume method has also been analyzed with finite element tools: 1] 4] 2] The convergence analysis has also been performed in some cases of nonlinear convection diffusion problems; see [19] with a combined finite element finite volume method, [14] and [17] with a pure finite volume scheme. A first order estimate for triangular meshes was obtained in [26] for a convection diffusion where the diffusion operator is the Laplacian under C 2 regularity assumptions of the solution. It generalizes easily to the case of Voronoi meshes, see [13] ....

..... 3 Discrete Poincar e inequalities and trace inequalities We give in this section some inequalities for piecewise constant functions. We recall here a discrete Poincar e inequality for the discrete H 1 0 norm of a piecewise constant function. The proof of this inequality may be found in [13] or [14]. Note that a discrete mean value Poincar e inequality may also be proven in order to deal with Neumann boundary conditions (see [13] Lemma 1 (Discrete Poincar e inequality) Let Omega be an open bounded polygonal subset of IR d , d = 2 or 3, T an admissible finite volume mesh in the sense ....

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Eymard R., T. Gallou et and R. Herbin , Convergence of finite volume schemes for semilinear convection diffusion equations, to appear in Numerische Mathematik.

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