A Diffuse Domain Approximation with Transmission-Type Boundary Conditions I: Asymptotic Analysis and Numerics

TL;DR

This paper analyzes a diffuse domain method for PDEs with transmission boundary conditions, demonstrating first-order asymptotic convergence and validating results through numerical simulations.

Contribution

It provides a matched asymptotic analysis of the diffuse domain method for transmission problems, establishing first-order accuracy in one dimension.

Findings

Asymptotic convergence to the original solution with first-order accuracy in ps

Numerical simulations confirm analytical predictions

Method effectively handles complex geometries with diffuse boundaries

Abstract

Diffuse domain methods (DDMs) have garnered significant attention for approximating solutions to partial differential equations on complex geometries. These methods implicitly represent the geometry by replacing the sharp boundary interface with a diffuse layer of thickness $\varepsilon$, which scales with the minimum grid size. This approach reformulates the original equations on an extended regular domain, incorporating boundary conditions through singular source terms. In this work, we conduct a matched asymptotic analysis of a DDM for a two-sided problem with transmission-type Robin boundary conditions. Our results show that, in the one dimensional space, the solution of the diffuse domain approximation asymptotically converges to the solution of the original problem, with exactly first-order accuracy in $\varepsilon$. Furthermore, we provide numerical simulations that validate and illustrate the analytical result.