A numerical framework for singular limits of a class of reaction diffusion problems

We present a numerical framework for solving localized pattern structures of reaction-diffusion type far from the Turing regime. We exploit asymptotic structure in a set of well established pattern formation problems to analyze a singular limit model that avoids time and space adaptation typically associated to full numerical simulations of the same problems. The singular model involves the motion of a curve on which one of the chemical species is concentrated. The curve motion is non-local with an integral equation that has a logarithmic singularity. We generalize our scheme for various reaction terms and show its robustness to other models with logarithmic singularity structures. One such model is the 2D Mullins-Sekerka flow which we implement as a test case of the method. We then analyze a specific model problem, the saturated Gierer-Meinhardt problem, where we demonstrate dynamic patterns for a variety of parameters and curve geometries.