Scalar auxiliary variable finite element scheme for the parabolic-parabolic Keller-Segel model

Alexandre Poulain (1, 2, 3)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 SU - Sorbonne Université
3 MAMBA - Modelling and Analysis for Medical and Biological Applications

We describe and analyze a finite element numerical scheme for the parabolic-parabolic Keller-Segel model. The scalar auxiliary variable method is used to retrieve the monotonic decay of the energy associated with the system at the discrete level. This method relies on the interpretation of the Keller-Segel model as a gradient flow. The resulting numerical scheme is efficient and easy to implement. We show the existence of a unique non-negative solution and that a modified discrete energy is obtained due to the use of the SAV method. We also prove the convergence of the discrete solutions to the ones of the weak form of the continuous Keller-Segel model.

Mots clés: Keller-Segel; Living tissues; Gradient flow; Energy stability

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Alexandre Poulain. Scalar auxiliary variable finite element scheme for the parabolic-parabolic Keller-Segel model. 2020. ⟨hal-02906142⟩