ON DISCRETE DUALITY FINITE VOLUME DISCRETIZATION OF GRADIENT AND DIVERGENCE OPERATORS IN 3D (original) (raw)

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.

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.

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Numerical Methods for Partial Differential Equations, 2007

Discrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo and Omnès for the Laplace equation, are proposed for nonlinear diffusion problems in 2D with nonhomogeneous Dirichlet boundary condition. This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of nonconformal locally refined meshes. We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general W−1,p′(Ω) source term and W1-(1/p),p(∂Ω) boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient, respectively, in appropriate Lebesgue spaces. Finally, error estimates are given in the case where the solution is assumed to be in W2,p(Ω). Numerical examples are given, including those on locally refined meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

A Finite Volume Scheme for Diffusion Problems on General Meshes Applying Monotony Constraints

SIAM Journal on Numerical Analysis, 2010

In order to increase the accuracy and the stability of a scheme dedicated to the approximation of diffusion operators on any type of grids, we propose a method which reduces the curvature of the discrete solution where the loss of monotony is observed. The discrete solution is shown to fulfill a variational formulation thanks to the use of Lagrange multipliers. We can then show its convergence to the solution of the continuous problem, and an error estimate is derived. A numerical method, based on Uzawa's algorithm, is shown to provide accurate and stable approximate solutions to various problems. Numerical results show the increase of precision due to the application of the method.

Finite volume schemes for the p-Laplacian on Cartesian meshes

ESAIM: Mathematical Modelling and Numerical Analysis, 2004

This paper is concerned with the finite volume approximation of the p-Laplacian equation with homogeneous Dirichlet boundary conditions on rectangular meshes. A reconstruction of the norm of the gradient on the mesh's interfaces is needed in order to discretize the p-Laplacian operator. We give a detailed description of the possible nine points schemes ensuring that the solution of the resulting finite dimensional nonlinear system exists and is unique. These schemes, called admissible, are locally conservative and in addition derive from the minimization of a strictly convexe and coercive discrete functional. The convergence rate is analyzed when the solution lies in W 2,p . Numerical results are given in order to compare different admissible and non-admissible schemes.

The Discrete Duality Finite Volume Method for Convection-diffusion Problems

SIAM Journal on Numerical Analysis, 2010

In this paper we extend the Discrete Duality Finite Volume (DDFV) formulation to the steady convection-diffusion equation. The discrete gradients defined in DDFV are used to define a cellbased gradient for the control volumes of both the primal and dual meshes, in order to achieve a higher-order accurate numerical flux for the convection term. A priori analysis is carried out to show convergence of the approximation and a global first-order convergence rate is derived. The theoretical results are confirmed by some numerical experiments.

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

The discretisation of the gradient operator is an important aspect of finite volume methods that has not received as much attention as the discretisation of other terms of partial differential equations. The most popular gradient schemes are the divergence theorem (or Green-Gauss) scheme, and the least-squares scheme. Both schemes are generally believed to be second-order accurate, but the present study shows that in fact the divergence theorem gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the least-squares gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. Furthermore, the schemes are used within a finite volume method to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the divergence theorem gradient scheme is inherited by the finite volume method as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the least-squares gradient leads to second-order accurate results. These numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM, which uses the divergence theorem gradient by default.

Analysis on general meshes of a discrete duality finite volume method for subsurface flow problems

This work presents and analyzes, on unstructured grids, a discrete duality finite volume method (DDFV method for short) for 2D-flow problems in nonhomogeneous anisotropic porous media. The derivation of a symmetric discrete problem is established. The existence and uniqueness of a solution to this discrete problem are shown via the positive definiteness of its associated matrix. Properties of this matrix combined with adequate assumptions on data allow to define a discrete energy norm. Stability and error estimate results are proven with respect to this norm. L 2 -error estimates follow from a discrete Poincaré inequality and an L ∞ -error estimate is given for a P 1 -DDFV solution. Numerical tests and comparison with other schemes (especially those from FVCA5 benchmark) are provided.