Robust Methods for Multiscale Coarse Approximations of Diffusion Models in Perforated Domains

TL;DR

This work develops robust multiscale coarse approximations for diffusion in perforated domains by constructing a high-order continuous Trefftz coarse space with piecewise-polynomial traces on a nonoverlapping coarse skeleton. It proves an $H^1$-projection error bound that depends on edge regularity and, with targeted edge refinement, yields superconvergence even for solutions with singularities. The coarse space is implemented discretely and embedded in a two-level domain decomposition framework (RAS), providing both an efficient iterative solver and a powerful preconditioner for Krylov methods, with strong scalability demonstrated on realistic urban geometries. Practically, the approach enables accurate, scalable diffusion computations in urban flood models and related multiscale problems, offering a systematic way to balance coarse-space size against convergence gains.

Abstract

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space which depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Combined with domain decomposition (DD) methods, the coarse space leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains. Numerical experiments showcase the increased precision of the coarse approximation as well as the efficiency and scalability of the coarse space as a component of a DD algorithm.

Figures (14)

• Figure 1: Finite element solution with $f = 1$ on model domains based on a smaller (left) and larger (right) data sets.
• Figure 2: Coarse (thick lines) and fine (thin lines) discretizations for smaller (left) and larger (right) data sets, with the coarse nodes shown by red dots.
• Figure 3: Coarse grid cell $\Omega_j$, nonoverlapping skeleton $\Gamma$ (blue lines), and coarse grid nodes $\mathbf{x}_s=(x_s,y_s) \in {\cal V}$ (red dots). Coarse grid nodes are located at $\overline{\Gamma} \cap \partial \Omega_S.$
• Figure 4: Coarse and fine discretizations of the L-shaped domain with 0 (left) and 1 (right) additional degree(s) of edge refinement.
• Figure 5: Coarse approximation error for L-shaped domain with edge (blue) and mesh (red) refinement in $L^2$ norm (left) and the energy norm (right). The black dashed line denotes the finite element error.

Theorems & Definitions (17)

• Lemma 2.1: Polynomial interpolation over an edge
• proof
• Lemma 2.2: Gagliardo-Nirenberg interpolation inequality
• proof
• Corollary 2.1: Interpolation in $H^s$
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof