Error analysis and convergence results for PDE discretization methods are normally based on the assumption, among other, that the PDE coefficients are continuous functions. When PDE discretization methods are applied to PDEs (BVPs or IVPs) with discontinuous coefficients, numerical results indicate that the standard convergence orders are typically not observed, and, even more, convergence is not guaranteed. We consider spline collocation PDE discretization methods and their application to a discontinuous diffusion PDE, modelling brain cancer growth, with the discontinuity of the diffusion coefficient arising from the different properties of the white and grey matters of the brain. We consider techniques based on adaptive grids and the approximation of the discontinuous coefficients by continuous ones, and techniques based on adjusting the basis functions so that they satisfy appropriate discontinuity conditions. We present numerical results highlighting the strengths and weaknesses of different approaches.