Meshfree Methods for Coupled Bulk-Surface Problems

In his 1952 paper “The Chemical Basis of Morphogenesis”, Alan Turing put forth a groundbreaking concept for developmental biology. He proposed that particular patterns seen in nature can be modelled as the steady state solution of some reaction-diffusion processes of two morphogens. This is nowadays commonly known as the Turing pattern, which can be found in real-life on the skin of an angelfish. Founded on what Turing originally outlined, reaction-diffusion models remain a subject undergoing intense study. In the recent past, mathematical models for cell biology were proposed to account for the chemical processes inside the cell (i.e., bulk domain) and on the cell membrane (i.e., surface). This project considers numerical methods for solving the resulting coupled bulk-surface systems of partial differential equations (PDEs).

The first challenging task is to come up with a sensible choice of trial spaces for solving coupled elliptic bulk-surface systems and provide a rigid convergence analysis for it. Coupled bulk-surface kernel-based collocation methods built upon some appropriate trial spaces would benefit from a convergence rate that is determined by the employed kernel and the smoothness of the solution but not the solutions of the coupled sys- tems. To define such a coupled trial space, we use the standard kernel-based trial space in the bulk domain. Besides our previously proposed embedded kernel-based meshfree collocation methods for elliptic PDEs on surfaces, we plan to develop and analyze an intrinsic version to compete against each other. The next objective is to provide easy- to-implement and efficient numerical methods for coupled bulk-surface reaction-diffusion models. First, we focus on a model system that couples diffusion in the bulk domain and reaction-diffusion on the surface. Individually, kernel-based collocation methods were proven to be highly successful on these problems. We then take what we learn from the model system to design algorithms in time-dependent settings. We measure success in terms of agreements between simulated results and the prediction made by the symmetry breaking instability theories in literatures. After identifying and verifying some suitable numerical formulations, we will start to address the issue of computational efficiency and adaptivity by studying bulk-surface reaction-diffusion systems, in which we have four unknown functions to solve.