A Finite Volume Scheme for Diffusion Problems on General Meshes Applying Monotony Constraints (original) (raw)

ON DISCRETE DUALITY FINITE VOLUME DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS IN 3D

2000

This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximations which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators. Following the approach developed by F. Hermeline and by K. Domelevo and P. Omnès, we consider a "double" covering T of a three-dimensional domain by a rather general primal mesh and by a well-chosen "dual" mesh. The associated discrete divergence operator div T is obtained by the standard finite volume approach. Then a consistent discrete gradient operator ∇ T is defined in such a way that −div T , ∇ T enjoy an analogue of the integration-by-parts formula known as the "discrete duality property". We discuss the implications of these properties and give a brief survey of other "discrete calculus" tools for "double" finite volume schemes.

On 3D DDFV discretization of gradient and divergence operators. I. Meshing, operators and discrete duality

IMA Journal of Numerical Analysis, 2012

This paper is the sequel of the paper [2] of S. Krell and the authors, where a family of 3D finite volume schemes on "double" meshes was constructed and the crucial discrete duality property was established. Heading towards applications, we state some discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discrete W 1,p compactness, discrete L 1 compactness in space and time) for the DDFV scheme of . We apply them to infer convergence of discretizations of nonlinear elliptic-parabolic problems of Leray-Lions kind. Applications to degenerate parabolic-hyperbolic PDEs and to a degenerate parabolic system known in electro-cardiology are briefly discussed.

Optimal discretization of smooth surfaces

2019

In der vorliegenden Arbeit untersuche ich die Diskretisierung glatter Flächen durch polyhedrale Flächen mit ebenen Facetten. Dabei soll die polyhedrale Fläche hinsichtlich ihrer Knickenergie minimiert werden. Hierbei ist die Knickenergie definiert als die Summe über alle Beträge der Winkel zwischen zwei angrenzenden Flächen, gewichtet mit der Länge der jeweils gemeinsamen Kante. Dieser Term lässt sich als diskrete Version einer totalen absoluten Krümmung interpretieren. Lässt man die Feinheit der Diskretisierung gegen Null gehen, so zeigt sich, dass ein Minimum der Knickenergie erreicht wird, wenn die Kanten entlang der Hauptkrümmungslinien der glatten Fläche verlaufen. Das lässt die Schlussfolgerung zu, dass eine polyhedrale Fläche mit minimaler Knickenergie aus ebenen Rechtecken bestehen sollte. Ein ähnliches Problem mit einer isotropen Version der Knickenergie wurde bereits in Martin Kilian (2017) behandelt. Der isotrope Fall lässt sich analog zum Volumenbeziehungsweise Kostenmin...

Discretization in 2D and 3D orders

Graphical Models, 2003

Among the different discretization schemes that have been proposed and studied in the literature, the supercover is a very natural one, and furthermore presents some interesting properties. On the other hand, an important structural property does not hold for the supercover in the classical framework: the supercover of a straight line (resp. a plane) is not a discrete curve (resp. surface) in general. We follow another approach based on a different, heterogenous discrete space which is an order, or a discrete topological space in the sense of Paul S. Alexandroff. Generalizing the supercover discretization scheme to such a space, we prove that the discretization of a plane in R 3 is a discrete surface, and we prove that the discretization of the boundary of any closed convex set X is equal to the boundary of the discretization of X .

A New Finite-Volume Approach to Efficient Discretization on Challenging Grids

SPE Journal, 2010

Summary Multipoint-flux-approximation (MPFA) methods were introduced to solve control-volume formulations on general simulation grids for porous-media flow. While these methods are general in the sense that they may be applied to any matching grid, their convergence properties vary. An important property for multiphase flow is the monotonicity of the numerical elliptic operator. In a recent paper (Nordbotten et al. 2007), conditions for monotonicity on quadrilateral grids have been developed. These conditions indicate that MPFA formulations that lead to smaller flux stencils are desirable for grids with high aspect ratios or severe skewness and for media with strong anisotropy or strong heterogeneity. The ideas were pursued recently in Aavatsmark et al. (2008), where the L-method was introduced for general media in 2D. For homogeneous media and uniform grids, this method has four-point flux stencils and seven-point cell stencils in two dimensions. The reduced stencils appear as a co...

Existence of second order discretizations on irregular mesh

Applied Mathematics Letters, 1989

We give the necessary and snfllcient conditions to obtain a difference formula of order two on an arbitrary distribution of mesh points. We also show how to form a 6 point computational cell in the physical plane if the mesh is generated by a boundary fitted coordinate generation procedure.

On 3D DDFV Discretization of Gradient and Divergence Operators: Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems

Computational Methods in Applied Mathematics, 2000

This paper is the sequel of the paper [2] of S. Krell and the authors, where a family of 3D finite volume schemes on "double" meshes was constructed and the crucial discrete duality property was established. Heading towards applications, we state some discrete functional analysis tools (consistency results, Poincaré and Sobolev embedding inequalities, discrete W 1,p compactness, discrete L 1 compactness in space and time) for the DDFV scheme of . We apply them to infer convergence of discretizations of nonlinear elliptic-parabolic problems of Leray-Lions kind. Applications to degenerate parabolic-hyperbolic PDEs and to a degenerate parabolic system known in electro-cardiology are briefly discussed.

The Gradient Discretisation Method

Mathématiques et Applications

HMM fluxes and link with the two-point finite volume method 13.4 A cell-centred variant of HMM schemes on ∆-admissible meshes364 13.5 The harmonic averaging points for heterogeneous domains. .. .

Notes on accuracy of finite-volume discretization schemes on irregular grids

Applied Numerical Mathematics, 2010

Truncation-error analysis is a reliable tool in predicting convergence rates of discretization errors on regular smooth grids. However, it is often misleading in application to finitevolume discretization schemes on irregular (e.g., unstructured) grids. Convergence of truncation errors severely degrades on general irregular grids; a design-order convergence can be achieved only on grids with a certain degree of geometric regularity. Such degradation of truncation-error convergence does not necessarily imply a lower-order convergence of discretization errors. In these notes, irregular-grid computations demonstrate that the design-order discretization-error convergence can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all.

On the finite volume element method

Numerische Mathematik, 1990

The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. This paper develops discretization error estimates for general self-adjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume.