Numerical analysis and simulations of a tractable model for tumour growth

This thesis begins with the description of a tractable model for tumour growth. The unique feature of this model is that we pass through the thin rim limit. We derive the sharp interface weak form and finite element scheme. We discuss the mesh smoothing techniques used in the implementation of the sharp interface finite element scheme. We then introduce an unfitted finite element scheme, and a sharp interface finite element scheme in R3. We also write the model in the diffuse interface paradigm, along with the associated weak form. We prove the existence and uniqueness of the solution to the di_use interface version of the model, and prove convergence of the diffuse interface finite element method. We conclude this thesis with a number of simulations in R2 and R3. Here, we present rates of convergence, and also investigate the effect of parameter spaces on the morphology of the tumour. A biologically motivated investigation is made, and a brief comparison with in vivo tumours is presented.