Projects per year
Abstract
A onespatial dimensional tumour growth model Breward et al. (2001, 2002, 2003) that consists of three dependent variables of space and time is considered. These variables are volume fraction of tumour cells, velocity of tumour cells, and nutrient concentration. The model variables satisfy a coupled system of semilinear advection equation (hyperbolic), simplified linear Stokes equation (elliptic), and semilinear diffusion equation (parabolic) with appropriate conditions on the timedependent boundary, which is governed by an ordinary differential equation. A reformulation of the model defined in a larger, fixed timespace domain is employed to overcome theoretical difficulties related to the timedependent boundary. This reformulation reduces the complexity of the model by removing the need to explicitly track the timedependent boundary. A numerical scheme that employs a finite volume method for the hyperbolic equation, a finite element method for the elliptic equation, and a backward Euler in timemass lumped finite element in space method for the parabolic equation is developed. We establish the existence of a time interval $(0,T_{\ast })$ over which a convergent subsequence of the numerical approximations can be extracted using compactness techniques. The limit of any such convergent subsequence is proved to be a weak solution of the continuous model in an appropriate sense, which is called a threshold solution. Numerical tests and justifications that support the theoretical findings conclude the paper.
Original language  English 

Pages (fromto)  11801230 
Number of pages  51 
Journal  IMA Journal of Numerical Analysis 
Volume  42 
Issue number  2 
DOIs  
Publication status  Published  Apr 2022 
Keywords
 convergence analysis
 finite elementfinite volume methods
 moving boundary
 tumour growth
 twophase models
Projects
 1 Finished

Discrete functional analysis: bridging pure and numerical mathematics
Droniou, J., Eymard, R. & Manzini, G.
Australian Research Council (ARC), Monash University, Université ParisEst Créteil Val de Marne (ParisEast Créteil University Val de Marne), University of California System
1/01/17 → 31/12/20
Project: Research