A vertex‐centered linearity‐preserving discretization of diffusion problems on polygonal meshes

Abstract

Summary This paper introduces a vertex‐centered linearity‐preserving finite volume scheme for the heterogeneous anisotropic diffusion equations on general polygonal meshes. The unknowns of this scheme are purely the values at the mesh vertices, and no auxiliary unknowns are utilized. The scheme is locally conservative with respect to the dual mesh, captures exactly the linear solutions, leads to a symmetric positive definite matrix, and yields a nine‐point stencil on structured quadrilateral meshes. The coercivity of the scheme is rigorously analyzed on arbitrary mesh size under some weak geometry assumptions. Also, the relation with the finite volume element method is discussed. Finally, some numerical tests show the optimal convergence rates for the discrete solution and flux on various mesh types and for various diffusion tensors. Copyright © 2015 John Wiley & Sons, Ltd.

New quadratic/serendipity finite volume element solutions on arbitrary triangular/quadrilateral meshes

“…Here we still solve the problem (1) and ( 2), and choose a discontinuous anisotropic diffusion coefficient as below Example 5. In the last example, we consider another strongly anisotropic diffusion problem which was investigated in [28]. The anisotropic diffusion tensor and exact solution are given by Λ(x, y) = 1 x 2 + y 2 Here we choose 𝜅 = 10 −3 .…”

“…Example In the last example, we consider another strongly anisotropic diffusion problem which was investigated in [28]. The anisotropic diffusion tensor and exact solution are given by $$$$ \Lambda \left(x,y\right)=\frac{1}{x^2+{y}^2}\left(\begin{array}{cc}\kappa {x}^2+{y}^2& \left(\kappa -1\right) xy\\ {}\left(\kappa -1\right) xy& {x}^2+\kappa {y}^2\end{array}\right),\kern1em u\left(x,y\right)=\mathit{\sin}\left(\pi x\right)\mathit{\sin}\left(\pi y\right), $$$$ where $$$ \kappa $$$ characterizes the level of anisotropy.…”

A novel vertex-centered finite volume method for solving Richards' equation and its adaptation to local mesh refinement

“…Our approach is based on a Vertex-Centered Finite Volume Method (hereinafter referred to as VCFVM) and sets all unknowns on grid vertices. It should be noted that the vertex-centered finite volume method has been widely studied for solving anisotropic diffusion/parabolic equations on general polygonal/polyhedral meshes in the past decade, see [41,40,45,36,5,29]. The numerical scheme proposed in this paper is partially inherited from [29], but substantial modifications in the spatial and temporal discretization have been made to consider the properties of Richards' equation, including the advection flux involved and the high nonlinearity between the soil hydraulic parameters and the soil water potential.…”