Difference Potentials Method for Models with Dynamic Boundary Conditions and Bulk-Surface Problems

TL;DR

This paper introduces a novel numerical framework based on Difference Potentials for solving 3D parabolic models with dynamic boundary conditions and bulk-surface coupling, applicable to biological and materials science problems.

Contribution

It develops new algorithms that efficiently handle bulk-surface coupling, irregular geometries, and spectral surface approximation using Cartesian meshes and Fast Poisson Solvers.

Findings

Algorithms demonstrate high accuracy and robustness.

Effective handling of complex 3D geometries.

Successful numerical tests validate the approach.

Abstract

In this work, we consider parabolic models with dynamic boundary conditions and parabolic bulk-surface problems in 3D. Such partial differential equations based models describe phenomena that happen both on the surface and in the bulk/domain. These problems may appear in many applications, ranging from cell dynamics in biology, to grain growth models in polycrystalline materials. Using Difference Potentials framework, we develop novel numerical algorithms for the approximation of the problems. The constructed algorithms efficiently and accurately handle the coupling of the models in the bulk and on the surface, approximate 3D irregular geometry in the bulk by the use of only Cartesian meshes, employ Fast Poisson Solvers, and utilize spectral approximation on the surface. Several numerical tests are given to illustrate the robustness of the developed numerical algorithms.