Analysis of the Parallel Schwarz Method for Growing Chains of Fixed-sized Subdomains : Part II

Abstract

According to classical theory, one level Schwarz methods applied to elliptic problems are not scalable in general [A. Toselli and O. Widlund, Springer Ser. Comput. Math., 34, Springer, New York, 2005]. This means that their convergence deteriorates when the number of subdomains increases. In contrast to this classical result, it was observed numerically in [E. Cancès, Y. Maday, and B. Stamm, J. Chem. Phys., 139 (2013), 054111; F. Lipparini, G. Scalmani, L. Lagardère, B. Stamm, E. Cancès, Y. Maday, J.-P. Piquemal, M. J. Frisch, and B. Mennucci, J. Chem. Phys., 141 (2014), 184108; F. Lipparini, B. Stamm, E. Cances, Y. Maday, and B. Mennucci, J. Chem. Theory Comput., 9 (2013), pp. 3637--3648] that in some cases the convergence of the one level Schwarz method does not depend on the number of subdomains. This happens for molecular problems where the domain of definition of the linear elliptic partial differential equation is the union of spherical van der Waal's cavities centered at the atomic position of the molecule. In this case, the computations can naturally be performed using Schwarz methods, where each atom of the molecule corresponds to a subdomain, see [E. Cancès, Y. Maday, and B. Stamm, J. Chem. Phys., 139 (2013), 054111; F. Lipparini, G. Scalmani, L. Lagardère, B. Stamm, E. Cancès, Y. Maday, J.-P. Piquemal, M. J. Frisch, and B. Mennucci, J. Chem. Phys., 141 (2014), 184108; F. Lipparini, B. Stamm, E. Cances, Y. Maday, and B. Mennucci, J. Chem. Theory Comput., 9 (2013), pp. 3637--3648]. We prove here that the scalability results presented in [G. Ciaramella and M. J. Gander, SIAM J. Numer. Anal., 55 (2017), pp. 1330--1356] for a simplified rectangular geometry also hold for realistic two-dimensional chains of circular subdomains. To do so, we first prove some characterization results for the solution of the Laplace equation in the unit disk. Then, using these combined with the maximum principle for harmonic functions, we obtain our convergence theorems for general configurations of molecules. Our convergence results reveal a further very unusual property of the Schwarz method in these simulations: starting from a certain critical overlap size, increasing the overlap further actually decreases the performance of the Schwarz method, in strong contrast to classical Schwarz theory.