|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
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.
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
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'
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.