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Key words: finite difference; finite element; partial differential equations; coupled systems; singularly perturbed problems; boundary and interior layers; numerical linear algebra; ADI; domain decomposition; multigrid methods.
Numerical Analysis of differential equations
I aim to design, analyse and implement robust methods for singularly perturbed problems. Solutions to these problems exhibit boundary and interior layers - narrow regions where the solution changes rapidly - and are known to be difficult to solve numerically. Robust methods should yield approximations whose accuracy is independent of the layer width. I have a particular interest in methods for coupled systems with solutions exhibiting multiple interacting layers of different scales, making their numerical resolution quite challenging.
Recent work includes the study of finite difference and finite element methods applied on specially design meshes for linearised convection-diffusion (=advection-diffusion) and reaction-diffusion problems.
Differential equations form one of the main languages in which mathematical models and be designed and expressed. Extracting useful information from these models often involves numerical simulation. While there are many excellent classical methods and black-box tools for this, the real world is rather complicated, and so too are the models of it. So one often needs bespoke numerical schemes. This is a fruitful source of interaction between computational mathematicians and applied scientists and engineers.
In recent times I've worked on the design of suitable methods for problems in wave-current interactions, dispersion of pollutants in coastal regions, and simulations of ICU patients reactions to clinical therapies.