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

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.

arXiv commentaccepted for publication in: Journal of Numerical Mathematics, 2022

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Cited by in corpus (4)

• Viscoelastic Cahn--Hilliard models for tumour growth
• Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions
• Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn-Hilliard systems with bounded mass source
• Activity-driven emulsification of phase-separating binary mixtures