Diffuse domain methods (DDMs) have attracted significant attention for approximating solutions to partial differential equations on complex geometries in two and three dimensions. These methods implicitly represent the geometry by replacing the sharp boundary interface with a diffuse layer of thickness $\epsilon$. We will define and consider both one-sided and two-sided versions of the problem. For the one-sided case, this approach reformulates the original equations on an extended, regular domain (for example, a hyper-cuboid). In both cases, boundary conditions are incorporated through nearly singular source terms.
In this work, we conduct a matched asymptotic analysis of a DDM for the two-sided problem with transmission-type boundary conditions. Our results show that, in one dimension, the solution of the diffuse domain approximation asymptotically converges to the solution of the original problem with exactly first-order accuracy in $\epsilon$. Furthermore, for the purely Neumann transmission boundary condition case, we show that the associated energy functional of the diffuse domain approximation Γ-converges to the energy functional of the original problem, and that the solution of the diffuse domain approximation strongly converges (up to a subsequence) to the solution of the original problem in $H^1(\Omega)$ as $\epsilon \to 0$.
The analysis of the DDM for the one-sided problem is much more subtle, as we will see. In this talk, I will discuss the theoretical analysis of the two-sided problem, in particular the details of the asymptotic and Γ-convergence analyses, and present numerical results that confirm the theory. I will conclude with some current work, including conjectures, on the more challenging one-sided problem.
325 Thackeray Hall