Abstract/Summary

The moving mesh finite element method (MMFEM) is a highly useful tool for the numerical solution of partial differential equations. In particular, for reaction-diffusion equations and multi-phase equations, the method provides the ability to track features of interest such as blow-up, the ability to track a free boundary, and the ability to model a dynamic interface between phases. This is achieved through a geometric conservation approach, whereby the integral of a suitable quantity is constant within a given patch of elements, but the footprint and location of those elements are dynamic. We apply the MMFEM to a variety of systems, including for the first time to various forms of the Lotka-Volterra competition equations. We derive a Lotka-Volterra based reaction-diffusion-aggregation system with two phases, representing spatially segregated species separated by a competitive interface. We model this system using the MMFEM, conserving an integral of population density within each patch of elements. We demonstrate its feasibility as a tool for ecological studies.