Analysis of a Schwarz-Fourier domain decomposition method

TL;DR

The paper analyzes Schwarz domain decomposition for the Laplace problem on a domain formed by two overlapping discs, focusing on an inexact Schwarz method where subproblems on each disc are solved using low-dimensional Fourier subspaces. By leveraging a variant of the maximum principle, it derives convergence results and contraction bounds for the Schwarz-Fourier iterations, including explicit positivity and boundary-layer considerations for the projected Poisson kernel. It introduces two algorithmic variants—Fourier-projection Schwarz-Fourier and Fourier-interpolation Schwarz-Fourier—establishing contraction bounds, computational efficiency via FFT, and robustness to small numbers of Fourier modes; numerical experiments illustrate the bounds and the influence of geometry and discretization on convergence. An appendix furnishes a rigorous endpoint analysis for Laplace solutions with boundary discontinuities, clarifying the limiting behavior at jump points and underpinning the maximum-principle arguments used in the main text.

Abstract

The Schwarz domain decomposition method can be used for approximately solving a Laplace equation on a domain formed by the union of two overlapping discs. We consider an inexact variant of this method in which the subproblems on the discs are solved approximately using the projection on a Fourier subspace of the $L^2$ space on the boundary. This model problem is relevant for better understanding of the ddCOSMO solver that is used in computational chemistry. We analyze convergence properties of this Schwarz-Fourier domain decomposition method. The analysis is based on maximum principle arguments. We derive a new variant of the maximum principle and contraction number bounds in the maximum norm.

Figures (10)

• Figure 2.1: Arcs $A_{\tilde{\theta}^\ast}$ on which the solution has constant value $u(\theta,r)= \frac{\tilde{\theta}^\ast - \theta^\ast}{\pi}$. \newlabelNGFig1
• Figure 3.1: Geometry of two overlapping discs $B_1=B((0,0);1)$ and $B_2=B((m,0);R)$. \newlabelFigGeom
• Figure 5.1: Projected kernels $K_N(\psi,r)$ for $N=5$ (left) and $N=25$ (right). \newlabelFigKN
• Figure 5.2: $1/\delta_{\rm num}(N)$ ($+$) and $1/\delta_{\rm th}(N)$ (curve); $q(N)=\frac{\delta_{\rm num}(N)}{\delta_{\rm th}(N)}$. \newlabelFigcompare
• Figure 5.3: $R_{\overline{\Gamma}_2} \Delta_1^{-1}P_N^1 \chi_{\Gamma_1}$, for $m=1.4$, $R=1.2$ (left) and $m=2.1$, $R=1.2$ (right). \newlabelFigExpPN

Theorems & Definitions (26)

• Example 2.1
• Lemma 2.1
• proof
• Remark 3.1
• Theorem 3.1
• proof
• Remark 3.2
• Remark 4.1
• Lemma 5.1
• proof