Macroscopic limit of diffuse domain approximations for porous media

Project description

Partial differential equations (PDEs) on complex domains (e.g., “swiss cheese” domains with holes or inclusions) are hugely relevant in the applied sciences, but they are notoriously difficult to solve numerically! Mimicking some concepts of a well-known mathematical description of fluid interfaces, known as diffuse-interface (or phase-field) model, the method of diffuse domains can reformulate PDEs on complex domains, imposing the boundary conditions weakly through a smoother indicator function defined on a homogeneous domain. This has several numerical and modelling advantages and it has been rigorously proved to be convergent to the solution on the exact (perforated) domains.

This project will explore this idea for heterogeneous materials and porous media, develop an implementation for advection-diffusion-reaction PDES, and study its macroscopic limit, when the inclusion diameters tend to zero, thereby providing a promising new approach to homogenisation of PDEs in complex domains.