Numerical solution of PDEs with discontinuous coefficients: application to brain cancer growth

Full Name: Christina Christara

Academic Affiliation: University of Toronto, Department of Computer Science

Position: Associate Professor

Abstract: Partial Differential Equations (PDEs) are ubiquitousin modelling various phenomena and situations. It is often impossible or undesirable to solve the PDEsarising from modelling by standard mathematical techniques.In such cases, numerical methods come into rescue. The PDE is discretized and the discrete problem is solved. A fundamental property that a numerical PDE method must exhibit is convergence of the discrete solution; and the higher the order of convergence, the better.  However, all standard 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 with discontinuous coefficients, numerical results indicate that the standard convergence orders are typically not observed, and, even more, convergence is not guaranteed.  We consider a particular PDE problem that models, through diffusion and reaction, the growth of tumoral cells in the brain, with the white and grey matters of the brain having different diffusion coefficients. We consider various numerical techniques to handle the discontinuous diffusion coefficient function, so that convergence (including high order) is exhibited.