Convergence of an implicit Voronoi finite volume method for reaction–diffusion problems

Abstract

We investigate the convergence of an implicit Voronoi finite volume method for reaction–diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical activities of the involved species as primary variables. The numerical scheme works with boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh‐independent global upper, and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. To illustrate the preservation of qualitative properties by the numerical scheme, we present a long‐term simulation of the Michaelis–Menten–Henri system. Especially, we investigate the decay properties of the relative free energy over several magnitudes of time, and obtain experimental orders of convergence for this quantity. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 141–174, 2016

Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations

“…It is now well understood that preserving at the discrete level the energy stability is of great importance for the accuracy in the long-time regime [18,19,35,36,6,37,2] or in some other asymptotic regime [7]. All these works are based on finite volumes with Two-Point Flux Approximation (TPFA) [40,29], and fail to extend to the anisotropic setting.…”