  We consider a one--spatial dimensional tumour growth model [2, 3, 4] that
consists of three dependent variables of space and time: 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. We employ a reformulation of the model defined in a larger, fixed
time-space domain to overcome some 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, but
nonlinearities in the equations, noncoercive operators in the simplified Stokes
equation, and interdependence between the unknown variables still challenge the
proof of suitable a priori estimates. 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 $(0,T_{\ast})$ over which, using compactness
techniques, we can extract a convergent subsequence of the numerical
approximations. The limit of any such convergent subsequence is proved to be a
weak solution of the continuous model in an appropriate sense, which we call a
threshold solution. Numerical tests and justifications that confirm the
theoretical findings conclude the paper.
