Analysis of the diffuse-domain method for solving PDEs in complex geometries

TL;DR

This paper analyzes the diffuse-domain method for solving PDEs in complex geometries, showing how boundary condition choices affect accuracy and proposing modifications for improved second-order accuracy.

Contribution

It provides a matched asymptotic analysis of the diffuse-domain method, identifying conditions for second-order accuracy and proposing correction terms and modifications.

Findings

Certain boundary condition approximations yield second-order accuracy in epsilon.

Other choices result in only first-order accuracy, affecting convergence.

Proposed modifications can achieve asymptotically second-order accuracy.

Abstract

In recent work, Li et al.\ (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness $\epsilon$ that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages. Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in $\epsilon$. However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the $L^2$ and $L^\infty$ norms for selected test problems.