Numerical analysis for a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport. (English) Zbl 1527.92002
J. Numer. Math. 30, No. 4, 295-324 (2022); corrigendum ibid. 32, No. 2, 213-214 (2024).
Summary: A diffuse interface model for tumour growth in the presence of a nutrient consumed by the tumour is considered. 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.
MSC:
| 92-08 | Computational methods for problems pertaining to biology |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 92C17 | Cell movement (chemotaxis, etc.) |
| 92C50 | Medical applications (general) |
Keywords:
Cahn-Hilliard equation; diffuse interface model; tumour growth; chemotaxis; numerical analysis; finite element method; simulationsReferences:
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