Dr. Marcus R Garvie

Overview of research areas:

Mathematical and numerical analysis

I am interested in the interdisciplinary approach of combining classical analysis, numerical analysis and scientific computing to investigate nonlinear partial differential equations (PDEs) in the applied sciences. My main areas of expertise are the rigorous application of the finite element method, and techniques from mathematical analysis to prove the global well-posedness of nonlinear PDEs. The methodology I use for the numerical analysis of PDEs is to mimic the properties of the continuous system in the discrete case. Thus mathematical analysis is used as a guide to practical computation. On the applied side I am interested in mathematical biology and nonlinear population dynamics in particular.

Mathematical biology

I am mainly concerned with PDEs that model applied problems, particularly in the biological sciences. An area that interests me is the application of mathematics to complex spatial pattern formation phenomena, for example: the development of mammalian coat patterns, epidemiology, plankton-dynamics, and the development of scroll-waves in cardiac tissue. Many pattern formation phenomena in biology can be modeled by nonlinear systems of reaction-diffusion equations, and thus numerical methods have an important part to play in finding practical solutions.

Doctoral work

During my doctoral studies I analyzed a system of coupled nonlinear reaction-diffusion equations of lambda-omega type that were originally devised in order to understand pattern formation in the Belousov-Zhabotinskii reaction (see for example J.D. Murray's classic book 'Mathematical Biology'). My doctoral work consisted of three parts: establishing the well-posedness of the PDEs using the Faedo-Galerkin method of J.L Lions (see also James Robinson's book 'Infinite-Dimensional Dynamical Systems'); numerical analysis using the 'lumped mass' finite element method with continuous piecewise linear basis functions; and scientific computing and simulation of spiral waves and 'target patterns' with Matlab and Fortran. My adviser was Dr. James Blowey. Used together these techniques provide a powerful and unified framework for the analysis of any nonlinear elliptic or parabolic system of PDEs in the applied sciences. For further details see my papers.


Last updated: Nov 16, 2006 by Marcus R Garvie