Numerical analysis for a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport

TL;DR

This paper develops and analyzes a finite element numerical scheme for a diffuse interface model of tumour growth involving Cahn-Hilliard and reaction-diffusion equations, proving stability, convergence, and providing numerical simulations.

Contribution

It introduces a fully-discrete finite element approximation for a tumour growth model with chemotaxis and active transport, establishing stability, existence, and convergence of solutions.

Findings

Proved stability bounds for the numerical scheme.

Established existence and continuous dependence of discrete solutions.

Demonstrated convergence to a global weak solution and provided numerical simulations.

Abstract

In this work, we consider a diffuse interface model for tumour growth in the presence of a nutrient which is consumed by the tumour. The system of equations consists of a Cahn--Hilliard equation with source terms for the tumour cells and a reaction-diffusion equation for the nutrient. We introduce a fully-discrete finite element approximation of the model and prove stability bounds for the discrete scheme. Moreover, we show that discrete solutions exist and depend continuously on the initial and boundary data. We then pass to the limit in the discretization parameters and prove convergence to a global-in-time weak solution to the model. Under additional assumptions, this weak solution is unique. Finally, we present some numerical results including numerical error investigation in one spatial dimension and some long time simulations in two and three spatial dimensions.