Finite Element Methods for Reaction-Diffusion Systems

The figure below represents the solution of the Fitzhugh-Nagumo equations:

The solution is characterized by the formation and expansion of a scroll wave that begins forming on the boundary of R₁ and spreads outward.

Solutions to such nonlinear partial differential equations are often obtained by discretization using the finite element method. Adaptive finite element methods seek to improve the accuracy and efficiency by increasing the discretization where errors are large and decreasing it where they are small. Thus, error estimates must be computed during the computation. These a posteriori error estimates and the adaptive machinery that utilizes them are the subjects of much of Professor Moore's research.