We present our finite element solutions to a continuum model of vascular tumour growth proposed by Hubbard and Byrne in [1]. Their model describes the evolution of a multiphase system of four incompressible phases – healthy cells, tumour cells, blood vessels and extracellular material – via three systems of partial differential equations (PDEs). These systems comprise: mass balance equations for the volume fraction of each phase; momentum balance equations for the velocity and pressure fields of each phase; and a reaction-diffusion system describing the transport and consumption of two essential nutrients, oxygen and glucose, that feed the growth of the tumour phase. We employ a carefully-selected combination of continuous and discontinuous finite element spaces to discretize the spatial domain, coupled with the three-stage strong-stability-preserving Runge-Kutta scheme for the time stepping., This scheme imposes the total variation-diminishing (TVD) property on the discontinuous solutions of the hyperbolic conservation laws which describe mass conservation. Our implementation is in python, making use of the Firedrake framework for low-level code generation [2]. This framework allows us to implement efficient solution and preconditioning technique by calling upon the PETSc toolkit [3]. The talk will begin with an overview of the continuum model and our discretization of it using appropriate finite element spaces. A range of numerical results will then be presented for discussion, before focusing on the main contribution of our work on the efficiency of the iterative methods used in the solution of the discretized systems of equations. In particular, we will demonstrate how well-designed preconditioners can significantly accelerate the linear algebra required at each time step – which is of enormous significance when seeking highly resolved simulations in three dimensions. The presentation will end with some proposals for future extensions of the model and our solutions schemes. REFERENCES [1] M. Hubbard and H. Byrne. Multiphase modelling of vascular tumour growth in two spatial dimensions. Journal of Theoretical Biology, 316(16):70-89, 2013. [2] “Firedrake.” [Online]. Available: https://www.firedrakeproject.org/. [3] E. Bueler. PETSc for Partial Differential Equations: Numerical Solutions in C and Python. Society for Industrial and Applied Mathematics, 2020.