Finite volume schemes for diffusion equations: introduction to and review of modern methods

We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature results, focusing on two important properties of schemes (discrete versions of well-known properties of the continuous equation): coercivity and minimum-maximum principles. Coercivity ensures the stability of the method as well as its convergence under assumptions compatible with real-world applications, whereas minimum-maximum principles are crucial in case of strong anisotropy to obtain physically meaningful approximate solutions

Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids

In this paper, we consider anisotropic diffusion with decay, and the diffusivity coefficient to be a second-order symmetric and positive definite tensor. It is well-known that this particular equation is a second-order elliptic equation, and satisfies a maximum principle under certain regularity assumptions. However, the finite element implementation of the classical Galerkin formulation for both anisotropic and isotropic diffusion with decay does not respect the maximum principle. We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with increase in the decay coefficient for isotropic medium and violates the discrete maximum principle. However, in the case of isotropic medium, the extent of violation decreases with mesh refinement. We then show that, in the case of anisotropic medium, the classical Galerkin formulation for anisotropic diffusion with decay violates the discrete maximum principle even at lower values of decay coefficient and does not vanish with mesh refinement. We then present a methodology for enforcing maximum principles under the classical Galerkin formulation for anisotropic diffusion with decay on general computational grids using optimization techniques. Representative numerical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation

On discrete functional inequalities for some finite volume schemes

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the space $BV(\Omega)$ into $L^{N/(N-1)}(\Omega)$ for a Lipschitz domain $\Omega \subset \mathbb{R}^{N}$, with $N \geq 2$. Finally, we give several applications to discrete duality finite volume (DDFV) schemes which are used for the approximation of nonlinear and non isotropic elliptic and parabolic problems