Abstract
This paper is concerned with the development and application of optimally efficient numerical methods for the simulation of vascular tumour growth. This model used involves the flow and interaction of four different, but coupled, phases which are each treated as incompressible fluids, Hubbard and Byrne (2013). A finite volume scheme is used to approximate mass conservation, with conforming finite element schemes to approximate momentum conservation and an associated equation. The principal contribution of this paper is the development of a novel block preconditioner for solving the linear systems arising from the discrete momentum equations at each time step. In particular, the preconditioned system has both a solution time and a memory requirement that is shown to scale almost linearly with the problem size.