Analysis of the Parallel Schwarz Method for Growing Chains of Fixed-Sized Subdomains: Part I

In implicit solvation models, the electrostatic contribution to the solvation energy can be estimated by solving a system of elliptic partial differential equations modeling the reaction potential. The domain of definition of such elliptic equations is the union of the van der Waals cavities corresponding to the atoms of the solute molecule. Therefore, the computations can naturally be performed using Schwarz methods, where each atom of the molecule corresponds to a subdomain. In contrast to classical Schwarz theory, it was observed numerically that the convergence of the Schwarz method in this case does not depend on the number of subdomains, even without coarse correction. We prove this observation by analyzing the Schwarz iteration matrices in Fourier space and evaluating corresponding norms in a simplified setting. In order to obtain our contraction results, we had to choose a specific iteration formulation, and we show that other formulations of the same algorithm can generate Schwarz iteration matrices with much larger norms leading to the failure of norm arguments, even though the spectral radii are identical. By introducing a new optimality concept for Schwarz iteration operators with respect to error estimation, we finally show how to find Schwarz iteration matrix formulations which permit such small norm estimates.