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

TL;DR

This paper introduces a finite element scheme using scalar auxiliary variables for the Keller-Segel model, ensuring energy decay, non-negativity, and convergence of solutions, thus improving numerical stability and efficiency.

Contribution

The paper develops a novel finite element scheme with scalar auxiliary variables that guarantees energy decay and solution convergence for the Keller-Segel model.

Findings

The scheme ensures monotonic energy decay at the discrete level.

Existence of a unique non-negative solution is proven.

Discrete solutions converge to the weak solution of the continuous model.

Abstract

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.