Lazarov R.D., I.D. Mishev and P.S Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal., 33, 1996, 31-55. 19  Home/Search   Document Details and Download   Summary   Related Articles   Check

....were carried out by [34] in the case of Cartesian meshes, 18] 3] 5] 6] and [38] in the case of unstructured meshes; see also [28] 36] 24] 29] and [35] in the case of quadrilateral meshes in two space dimensions. Cell centered finite volume schemes are addressed in [25] 15] 41] and [23] in the case of Cartesian meshes and in [40] 19] 20] 22] 26] in the case of triangular (in two space dimensions) or Voronoi meshes; let us also mention [8] and [9] where more general meshes are treated, with, however, a somewhat technical geometrical condition. In the pure diffusion case, ....

Lazarov R.D., I.D. Mishev and P.S Vassilevski (1996), Finite volume methods for convectiondiffusion problems, SIAM J. Numer. Anal., 33, 1996, 31-55.

.... attempts have been made to generalize it to two dimensions: finite element methods that use approximate L splines (i.e. trial functions that lie locally in the null space of an approximation of L) 10, 11, 12, 16, 17] mixed finite element methods [4] finite volume methods (and the box method) [3, 9, 13] and exponentially fitted finite difference schemes [8, 15] are some of the generalizations suggested in the literature. In the present paper, we shall work with finite element methods, using a trial space of approximate L splines that was recently devised in [17] We first discuss the ....

R. D. Lazarov, I. D. Mishev & P. S. Vassilevski, Finite volume methods for convection-diffusion problems, to appear in SIAM J. Numer. Anal. (1995)

....term, the scheme can also be interpreted as of finite volume type. Among the recent related works dealing with the convergence and accuracy properties of finite volume schemes, we can mention the error estimates obtained by a variational approach; in such a context both discrete (e.g. [5, 22]) and continuous (e.g. 33] error estimates are known. Concerning the finite volume analysis for hyperbolic conservation laws in the unstructured case, we can refer to Vila [39] for an error estimate in L 1( Omega Gamma and to Champier, Gallouet [7] and Cockburn et al. 10] for convergence ....

R. D. LAZAROV, I. D. MISHEV and P. S. VASSILEVSKI. Finite volume methods for convection-diffusion problems. To appear.

....difference approximation of the fluxes at the interfaces. Recently, error estimates and convergence results for the cell centered finite volume approximations on structured or unstructured 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 ....

R.D. Lazarov, I.D. Mishev and P.S Vassilevski, Finite volume methods for convectiondiffusion problems, SIAM J. Numer. Anal., 33, 31-55, 1996.

....first one consists in writing the finite volume as a finite element or mixed finite element method by using some numerical integration, see for instance [1] 2] 17] 18] or [19] the convergence then follows from the general finite element framework. The second one, see for example [5] 6] [11], 9] 20] 13] or [21] consists in establishing the convergence by using the direct formulation of the finite volume scheme together with some appropriate discrete functional analysis tools. This last approach is considered here. A discrete system is obtained for each type of boundary ....

....in four sections. The first one introduces the admissible meshes which are needed for the discretization of the elliptic problem, and the three following sections correspond to the three types of boundary conditions which we consider here. Homogeneous Dirichlet conditions were studied in e.g. [11], 20] 9] 13] with different assumptions on the data and the mesh; to our knowledge, nonhomogeneous Dirichlet, Neumann and Robin boundary conditions have only been considered up to now in [6] with some simplifying assumptions; the convergence of the method for Neumann and Robin conditions ....

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R.D. Lazarov, I.D. Mishev and P.S. Vassilevski (1996), Finite volume methods for convection diffusion problems, SIAM J. Numer. Anal. 33, 31-55.

....first one consists in writing the finite volume as a finite element or mixed finite element method by using some numerical integration, see for instance [1] 2] 18] 19] or [20] the convergence then follows from the general finite element framework. The second one, see for example [5] 6] [12], 9] 21] 14] or [22] consists in establishing the convergence by using the direct formulation of the finite volume scheme together with some appropriate discrete functional analysis tools. This last approach is considered here. A discrete system is obtained for each type of boundary ....

....in four sections. The first one introduces the admissible meshes which are needed for the discretization of the elliptic problem, and the three following sections correspond to the three types of boundary conditions which we consider here. Homogeneous Dirichlet conditions were studied in e.g. [12], 21] 9] 14] with different assumptions on the data and the mesh; to our knowledge, nonhomogeneous Dirichlet, Neumann and Robin boundary conditions have only been considered up to now in [6] with some simplifying assumptions; the convergence of the method for Neumann and Robin conditions ....

[Article contains additional citation context not shown here]

R.D. Lazarov, I.D. Mishev and P.S. Vassilevski (1996), Finite volume methods for convection diffusion problems, SIAM J. Numer. Anal. 33, 31-55.

....finite volume method, and to [3] 4] 14] and [11] for the related finite volume element and control volume finite element methods. The analysis of cell centered finite volume schemes has only recently been undertaken. Error estimates were first obtained in the rectangular case [23] 15] [22]. Triangular meshes and Voronoi meshes, which we shall also refer to as admissible meshes, were also investigated [27] 18] 12] 21] convergence results were obtained for Dirichlet boundary conditions and constant diffusion coefficients and were generalized to Neumann and Fourier boundary ....

Lazarov R.D., I.D. Mishev and P.S Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal., 33, 1996, 31-55.

.... see [11] to the case of a diffusion operator involving discontinuous tensor diffusion coefficients and the time dependent case [19] Error estimates assuming H 2 regularity of the solution may also be obtained for linear convection diffusion equations for Dirichlet boundary conditions [21] [22], 24] 11] and Neumann or Fourier boundary conditions [11] Note also that the finite volume scheme is well adapted to the discretization of hyperbolic systems (see e.g. 11] and references therein) and is therefore a good candidate for the discretization of systems of equations of different ....

Lazarov R.D., I.D. Mishev and P.S Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal., 33, 1996, 31-55.

....meshes; see also [36] 43] and [33] in the case of quadrilateral meshes. We are interested here by the so called cell centered approach, i.e. the discrete unknowns are located at some point in the control volumes. Cell centered finite volume schemes are addressed in [34] 23] 50] and [30] in the case of Cartesian meshes and in [47] 26] 27] 29] 35] in the case of triangular or Voronoi meshes; let us also mention [9] and [7] where more general meshes are treated, with, however, a somewhat technical geometrical condition. In the pure diffusion case, the cell centered finite ....

.... see [13] to the case of a diffusion operator involving discontinuous tensor diffusion coefficients and the time dependent case [27] Error estimates assuming H 2 regularity of the solution may also be obtained for linear convection diffusion equations for Dirichlet boundary conditions [29] [30], 32] 13] and Neumann or Fourier boundary conditions [13] Note also that the finite volume scheme is well adapted to the discretization of hyperbolic systems (see e.g. 13] and references therein) and is therefore a good candidate for the discretization of systems of equations of different ....

Lazarov R.D., I.D. Mishev and P.S Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal., 33, 1996, 31-55.

.... 14] Wheeler [55] and in references cited therein and by modern FV error analysis for the lowest order methods based on FE style arguments for linear finite elements: e.g. Bank and Rose [5] Ewing et al. 17, 18] Hackbusch [21] Heinrich [23] Herbin [24] Kreiss et al. 31] Lazarov et al. [32, 33], Manteuffel and White [35] Morton and Suli [38] Samarskii et al. 46] Suli [50] and Weiser and Wheeler [54] By adapting FV and FE arguments to fit the context of FVE and developing some new arguments unique to FVE, we derive optimal order error estimates under the full range of admissible ....

....(see Mitchell and Griffiths [36] Kreiss et al. 31] Manteuffel and White [35] and the references listed therein) FV is viewed as a special type of finite difference method and analyzed in terms of finite difference truncation error in C 1 norms. However, in modern finite volume analysis (see [5, 17, 18, 21, 23, 32, 33, 38, 46, 50, 54]) FV is viewed as a special type of finite element method and its error is analyzed in terms of finite element approximation theory results: results for the error in approximating elements of a fractional order Sobolev space with piecewise polynomials. Since FVE is even closer to FE than FV, our ....

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R. D. Lazarov, I. D. Mishev, and P. S. Vassilevski, "Finite volume methods for convection-diffusion problems," SIAM J. Numer. Anal., 33 (1996), pp. 31--55.

....are not optimal with respect of the regularity of the solution. To certain extend this deficiency has been overcome by Samarskii, Lazarov, and Makarov in [61] for diffusion reaction equations. Further, various unconditionally stable schemes have been studied by Lazarov, Mishev, and Vassilevski in [44] where optimal with respect to the regularity of the solution error estimates have been established. Approximations on locally refined rectangular and triangular grids have been studied in [31, 32, 33, 66] Extensions of the finite difference approximations to non rectangular grids have been ....

....is a consequence of the construction of the method. For piece wise constant coefficients the proof uses the equivalence of the finite volume approximation to the finite element approximation (see, e.g. 40] For rectangular grids a detailed proof (including convergence in L 2 norm) is given in [44]. Similarly to the finite element method the discrete problem is unconditionally stable in H norm. Moreover, the above approximation of the convection part of the operator produces an M matrix. Therefore, if the diffusion part produces also an M matrix (for example this will be the case if ....

R. D. Lazarov, I. D. Mishev, and P. S. Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal. 33, 1 (1996), 31--55.

....1 and 2 clearly show second order accuracy in L 2 norm and first order inH 1 norm. We observe some fluctuations that can be explained with geometrical irregularities or and not exact decreasing of the area in consecutive triangulations mention above. Compare with the results reported in [19]) We point out that our theory for convection dominated case have to be considered as asymptotic, i.e. for sufficiently small h. Results reported for problems 3 and 4 show first order in L 2 norm. For a = 10 Gamma2 the rate of convergence in H 1 norm increases with refinement of the ....

R. D. Lazarov, I. D. Mishev, and P. S. Vassilevski. Finite volume methods for convection-diffusion problems. SIAM J.Numer. Anal., 33(1):31--55, 1996.

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Lazarov R.D., I.D. Mishev and P.S Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal., 33, 1996, 31-55. 19

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