Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations

Abstract

In this paper, we propose and analyze a Control Volume Finite Elements (CVFE) scheme for solving possibly degenerated parabolic equations. This scheme does not require the introduction of the so-called Kirchhoff transform in its definition. We prove that the discrete solution obtained via the scheme remains in the physical range, and that the natural entropy of the problem decreases with time. The convergence of the method is proved as the discretization steps tend to 0. Finally, numerical examples illustrate the efficiency of the method.

Positive nonlinear CVFE scheme for degenerate anisotropic Keller-Segel system

“…According to [8,9], we approximate the degenerate diffusion term in its original form in (1.1). Next, we use the specific form of the chemoattractant function to propose a new scheme preserving the positivity of solutions and convergent.…”

Entropy-Diminishing CVFE Scheme for Solving Anisotropic Degenerate Diffusion Equations

“…Hence, it preserves the natural L ∞ bounds 0 and 1. This is the purpose of Proposition 3 (the proof is given in [2]), that moreover ensures that all the terms in (6) are finite.…”

“…[7]), we aim to discretize the problem in its from (1) rather than in its form (2). Moreover, we aim to derive a method such that the L ∞ -estimate (3) remains true at the discrete level despite the anisotropy of the problem, such that the discrete counterpart of the entropy Ω H(u(x,t))dx decreases with time as prescribed by (4) in the continuous setting, and such that the discrete solution converges towards the unique weak solution as the discretization steps tend to 0.…”

“…All the numerical analysis results stated in this section are thoroughly justified in the forthcoming paper [2]. First, let us give some a priori estimates.…”

“…The convergence of the discrete solution u M ,∆t as the space and time discretization steps tend to 0 towards the unique weak solution u of the continuous problem is the purpose of the following theorem, whose proof is contained in [2]. The proof of Theorem 1 follows (with some additional technical difficulties) the path proposed in [4], that consists in first proving some compactness on the family of discrete solutions u M m ,∆t m m≥1 , and then to identify any limit value (up to a subsequence) u = lim m→∞ u M m ,∆t m as the unique weak solution to the problem (2).…”