On 3D DDFV Discretization of Gradient and Divergence Operators: Discrete Functional Analysis Tools and Applications to Degenerate Parabolic Problems (original) (raw)
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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 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.
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.
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
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.
In this chapter is exposed a methodology for the Conventional Finite Volume analysis of steady state flow problems governed by discontinuous diffusion coefficients. This methodology combines a rigorous mathematical approach with the respect of physical principles for fluid flows in porous media (Principle of pressure and flux continuity across grid-block boundaries and Principle of Mass conservation for each control-volume, for instance). Taking the incompressible one-phase flow in heterogeneous porous media as a reference model, a great attention is put on the equivalent (or homogenized) absolute permeability involved in the expression of the discrete Darcy velocity over the "interaction zone" between two adjacent mesh elements. The first key-step of our methodology consists in putting in place a very general discrete-function-space framework with the corresponding inner products and their associated norms. Then after adequate mathematical tools are deployed as projection and interpolation operators with their fundamental properties. A discrete version of the Poincaré-Friedrichs inequality is also established and used later to prove the coerciveness of the bilinear form involved in the discrete variational formulation of the model problem when homogeneous Dirichlet boundary conditions are (partially or fully) prescribed. Indeed techniques similar to those of Finite Element Methods have been used for: (i) Getting a discrete variational setting of the model problem, (ii) Proving existence and uniqueness via the Lax-Milgram theorem, (iii) Showing stability of the discrete solution (thanks to continuity property of Projection Operators introduced before). A novel technique based upon geometric arguments is deployed to prove that the discrete solution meets the discrete maximum principle in the context of homogeneous Dirichlet boundary conditions. Interpolation Operators are used to define cell-wise constant and linear spline approximate solutions. A first order convergence in L 2 −norm and in some discrete energy norm has been exposed (see Lemma 2.36). Sufficient Conditions to get higher order convergence rate in L 2 −norm and in H 1 0 −norm have been stated (see Proposition 2.47).
IMA Journal of Numerical Analysis, 2011
We study the numerical approximation to the solution of the steady convection-diffusion equation. The diffusion term is discretized by using the hybrid mimetic method (HMM), which is the unified formulation for the hybrid finite-volume (FV) method, the mixed FV method and the mimetic finite-difference method recently proposed in Droniou et al. (2010, Math. Models Methods Appl. Sci., 20, 265-295). In such a setting we discuss several techniques to discretize the convection term that are mainly adapted from the literature on FV or FV schemes. For this family of schemes we provide a full proof of convergence under very general regularity conditions of the solution field and derive an error estimate when the scalar solution is in H 2 (Ω). Finally, we compare the performance of these schemes on a set of test cases selected from the literature in order to document the accuracy of the numerical approximation in both diffusion-and convection-dominated regimes. Moreover, we numerically investigate the behaviour of these methods in the approximation of solutions with boundary layers or internal regions with strong gradients.
Journal of Applied Mathematics, 2016
A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms and 2-norm), are investigated. Numerical test is provided.