Trace estimates for harmonic functions along circular arcs with applications to domain decomposition on overlapping disks

Abstract

In this paper we derive several (and in many cases sharp) estimates for the $\mathrm{L}^2$-trace norm of harmonic functions along circular arcs. More precisely, we obtain geometry-dependent estimates on the norm, spectral radius, and numerical range of the Dirchlet-to-Dirichlet (DtD) operator sending data on the boundary of the disk to the restriction of its harmonic extension along circular arcs inside the disk. The estimates we derive here have applications in the convergence analysis of the Schwarz domain decomposition method for overlapping disks in two dimensions. In particular, they allow us to establish a rigorous convergence proof for the discrete parallel Schwarz method applied to the Conductor-like Screening Model (COSMO) from theoretical chemistry in the two-disk case, and to derive error estimates with respect to the discretization parameter, the number of Schwarz iterations, and the geometry of the domain. Our analysis addresses challenges beyond classical domain decomposition theory, especially the weak enforcement of boundary conditions.