Mishev ID. Finite volume methods on Voronoi meshes. Numerical Methods for Partial Di#erential Equations 1998; 14:193--212.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Department of Mathematics, University of Alberta, Edmonton, AL T6G 2G1 Canada (ylin hilbert.math.ualberta.ca) 1865 are desirable in many applications. However, the analysis of FV methods lags far behind that of finite element and finite di#erence methods. Readers are referred to [3, 6, 17, 21, 22, 25] for some recent developments. The FVE method considered in this paper is a variation of the FV method, which also can be considered as a Petrov Galerkin finite element method. Much has been published on the accuracy of FVE methods using conforming linear finite elements. Some early work ....

.... but its regularity requirement on the exact solution seems to be too high compared with that for finite element methods having an optimal order convergence rate when the exact solution is in W 2(##2 Optimal order H estimates and superconvergence in a discrete H norm also have been given in [3, 17, 21, 22, 25] under various assumptions on the above form for equations or triangulations. More recently, the authors of [7, 8] presented a framework based on functional analysis to analyze the FVE approximations. The authors in [11] obtained some new error estimates by extending the techniques of [20] The ....

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I.D. Mishev, Finite volume methods on Voronoi meshes, Numer. Methods Partial Di#erential Equations, 14 (1998), pp. 193--212.

....of the convection term is closely related to the discontinuous Galerkin approximation (see, e.g. the survey paper by Arnold, Brezzi, Cockburn, and Marini [4] or to the Tabata scheme for Galerkin finite element method [64] Remark 1. Voronoi meshes have some advantages in 2 D (see, e.g. [54]) A different type of weighted upwind approximation on Voronoi meshes in 2 D has been studied in [2] However, these meshes are not well suited for adaptive grid refinement and their generalization to 3 D problems is not immediate or simple. Remark 2. Further applications of this method to ....

I. D. Mishev, Finite volume methods on Voronoi meshes, Numer. Methods PDEs 14 (1998), 193--212.

....of the edges of T . These types of control volumes form the so called Voronoi grid. Then obviously, fl ij are the perpendicular bisectors of the three edges of T (see Figure 2) This requires that all elements are triangles of acute type, which we shall assume whenever such a grid is used. See [1 3,8,10,11] for some detail analyses and historical developments of finite volume element methods. 8 x x j V i g ij x ij x x V g i i ij ij x Fig. 1. Control volumes with medicenters as internal points and interface fl ij of V i and V j . p q K x x j x ij V i g ij Fig. 2. Control volumes with ....

I.D. Mishev, Finite Volume Methods on Voronoi Meshes, Numerical Methods for Partial Differential Equations 14 (1998), 193--212. 18

.... X ffi;k be the piecewise constant interpolation operator, that is I hk u = X x i;k 2Nh k u i;k i;k (x) where u i;k = u(x i;k ) Then we set I ffi = Q K k=1 I hk and I ffi = Q K k=1 I hk . With the above preparation, we can combine the finite volume approximation (see, e.g. [10, 14, 15, 16]) with 5 the mortar approach to define our mortar finite volume element method: find (u ffi ; ffi ) 2 X ffi Theta M ffi such that A(u ffi ; I ffi v ffi ) B(v ffi ; ffi ) f; I ffi v ffi ) v ffi 2 X ffi ; B(u ffi ; OE ffi ) 0; OE ffi 2 M ffi ; 5) where A(u ffi ; I ffi v ....

I. D. Mishev, Finite volume methods on Voronoi meshes, Numer. Meth. for PDEs, v. 14 (1998), 193-212. 8

....certain quantity (mass, heat, momentum, etc) over each in nitesimal volume. The nite volume method has been combined with the technique of the nite element method in a new development which is capable of producing accurate approximations on general simplicial and quadrilateral grids (see, e.g. [4, 5, 6, 7, 8, 13]) For a collection of theoretical results and various applications we refer [2] The main advantages of the nite volume method are compactness of the discretization stencil, good accuracy, and discrete local conservation, which for many applications is a very 2 desirable feature of the ....

I.D. Mishev, Finite volume methods on Voronoi meshes, Numerical Methods for Partial Dierential Equations, 14 (1998), 193-212.

....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 spaces which we ....

I.D. Mishev, Finite volume methods on Voronoi meshes, Num. Meth. P.D.E. 14, 2, 193-212, 1998.

....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 condition. Existence and ....

....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 requires some ....

I. D. Mishev (1998), Finite volume methods on Voronoi meshes, Num. Meth. P.D.E. 14, 2, 193-212.

.... each nite element T , then the mortar nite volume element method introduced above coincides with the the standard mortar Galerkin nite element method with piecewise linear elements studied, for example, in [7, 10, 11] This is due to the fact, established by Jianguo and Shitong [34] see, also [21, 23, 37, 38]) Lemma 2.2. see, 34, 38] Assume that the matrix A(x) is constant on each T 2 T k for k = 1; K. Then A(u ; v ) A(u ; I v ) 8u ; v 2 X : 2.9) Therefore, for piecewise constant coecients, the mortar nite element and nite volume approximations produce the same element ....

....nite volume element method introduced above coincides with the the standard mortar Galerkin nite element method with piecewise linear elements studied, for example, in [7, 10, 11] This is due to the fact, established by Jianguo and Shitong [34] see, also [21, 23, 37, 38] Lemma 2.2. see, [34, 38]) Assume that the matrix A(x) is constant on each T 2 T k for k = 1; K. Then A(u ; v ) A(u ; I v ) 8u ; v 2 X : 2.9) Therefore, for piecewise constant coecients, the mortar nite element and nite volume approximations produce the same element sti ness matrices. The ....

[Article contains additional citation context not shown here]

I.D. Mishev, Finite Volume Methods on Voronoi Meshes, Numerical Methods for Partial Dierential Equations, 14 (1998), 193-212.

....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 condition. Existence and ....

....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 requires some ....

I. D. Mishev (1998), Finite volume methods on Voronoi meshes, Num. Meth. P.D.E. 14, 2, 193-212.

.... [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 types, ....

I.D. Mishev, Finite volume methods on Voronoi meshes, to appear.

....is adjoint in a special inner product to the discrete divergence. In recent years, the finite volume approach has been combined with finite element method techniques in a new development which is capable of producing accurate approximations on general triangular and quadrilateral grids (see, e.g. [2, 3, 4, 11, 13, 16]) The main advantages of the method are compactness of the discretization stencil, good accuracy, and local discrete conservation. In all of these discretization methods, it is assumed that the possible jumps of the diffusion coefficient are aligned with the finite element partitioning. This ....

I.D. Mishev, Finite Volume Methods on Voronoi Meshes, Numer. Meth. for PDEs, 14 (1998), 193--212.

.... [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 types, ....

I.D. Mishev, Finite volume methods on Voronoi meshes, to appear.

No context found.

Mishev ID. Finite volume methods on Voronoi meshes. Numerical Methods for Partial Di#erential Equations 1998; 14:193--212.

No context found.

I.D. Mishev, Finite Volume Methods on Voronoi Meshes, Numer. Meth. for PDEs, 14 (1998), 193--212.

No context found.

I.D. Mishev, Finite volume methods on Voronoi meshes, to appear.

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