A Combined Finite Volume-Finite Element Scheme for the Discretization of Strongly Nonlinear Convection-Diffusion-Reaction Problems on Nonmatching Grids

作者：Eymard Robert; Hilhorst Danielle; Vohralik Martin^*

来源：Numerical Methods for Partial Differential Equations, 2010, 26(3): 612-646.

DOI：10.1002/num.20449

• degenerate parabolic convection-diffusion-reaction equation
• inhomogeneous and anisotropic diffusion
• convection dominance
• nonmatching grids
• finite volume method
• finite element method
• harmonic averaging
• local Peclet upstream weighting
• convergence of approximate solutions

We propose and analyze in this paper a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and source terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell-centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the conforming piecewise linear finite element method. In this way, the scheme is fully consistent and the discrete solution is naturally continuous across the interfaces between the subdomains with nonmatching grids, without introducing any supplementary equations and unknowns or using any interpolation at the interfaces. We allow for general inhomogeneous and anisotropic diffusion dispersion tensors, propose two variants corresponding respectively to arithmetic and harmonic averaging, and use the local Peclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection-dominated case. The scheme is robust, efficient since it leads to positive definite matrices and one unknown per element, locally conservative, and satisfies the discrete maximum principle under the conditions on the simplicial mesh and the diffusion tensor usual in the finite element method. We prove its convergence using a priori estimates and the Kolmogorov relative compactness theorem and illustrate its behavior on a numerical experiment.

• 出版日期2010-5