Convergence analysis of a numerical scheme for a tumour growth model

Abstract

A one–spatial 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 time–dependent boundary, which is governed by an ordinary differential equation. A reformulation of the model defined in a larger, fixed time–space domain is employed to overcome theoretical difficulties related to the time–dependent boundary. This reformulation reduces the complexity of the model by removing the need to explicitly track the time–dependent 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 time–mass lumped finite element in space method for the parabolic equation is developed. We establish the existence of a time interval (0,T∗) 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.