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

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 twodimensional 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.

[1]  Thanh N. Truong,et al.  A new method for incorporating solvent effect into the classical, ab initio molecular orbital and density functional theory frameworks for arbitrary shape cavity , 1995 .

[2]  Martin J. Gander,et al.  Optimized Schwarz methods with nonoverlapping circular domain decomposition , 2016, Math. Comput..

[3]  Zi-Cai Li,et al.  Schwarz Alternating Method , 1998 .

[4]  Martin J. Gander,et al.  Analysis of the Parallel Schwarz Method for Growing Chains of Fixed-Sized Subdomains: Part I , 2017, SIAM J. Numer. Anal..

[5]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[6]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[7]  Benjamin Stamm,et al.  Domain decomposition for implicit solvation models. , 2013, The Journal of chemical physics.

[8]  Martin J. Gander,et al.  Can the discretization modify the performance of Schwarz methods , 2011 .

[9]  Block Jacobi Relaxation for Plane Wave Discontinuous Galerkin Methods , 2014 .

[10]  Stephen J. Gardiner,et al.  Classical Potential Theory , 2000 .

[11]  Martin J. Gander.  On the influence of geometry on optimized Schwarz methods , 2011 .

[12]  Benjamin Stamm,et al.  Quantum, classical, and hybrid QM/MM calculations in solution: general implementation of the ddCOSMO linear scaling strategy. , 2014, The Journal of chemical physics.

[13]  Martin J. Gander,et al.  Continuous analysis of the additive Schwarz method: A stable decomposition in H1 with explicit constants , 2015 .

[14]  Martin J. Gander,et al.  Non Shape Regular Domain Decompositions: An Analysis Using a Stable Decomposition in , 2013, Domain Decomposition Methods in Science and Engineering XX.

[15]  Martin J. Gander,et al.  Block Jacobi for Discontinuous Galerkin Discretizations: No Ordinary Schwarz Methods , 2014 .

[16]  P. G. Ciarlet,et al.  Linear and Nonlinear Functional Analysis with Applications , 2013 .

[17]  Benjamin Stamm,et al.  Fast Domain Decomposition Algorithm for Continuum Solvation Models: Energy and First Derivatives. , 2013, Journal of chemical theory and computation.

[18]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[19]  Peter J. Olver,et al.  Complex Analysis and Conformal Mapping , 2015 .

[20]  M. Peloso.  CLASSICAL SPACES OF HOLOMORPHIC FUNCTIONS , 2015 .

[21]  Daniel J. Arrigo,et al.  An Introduction to Partial Differential Equations , 2017, An Introduction to Partial Differential Equations.

[22]  V. Barone,et al.  Quantum Calculation of Molecular Energies and Energy Gradients in Solution by a Conductor Solvent Model , 1998 .

[23]  Martin J. Gander,et al.  Optimized Schwarz Method with Two-Sided Transmission Conditions in an Unsymmetric Domain Decomposition , 2016 .

[24]  A. Klamt,et al.  COSMO : a new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient , 1993 .

[25]  Martin J. Gander,et al.  Optimized Schwarz Methods , 2006, SIAM J. Numer. Anal..

[26]  Martin J. Gander,et al.  Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization: The many-subdomain case , 2016, Math. Comput..

[27]  Martin J. Gander,et al.  On the Influence of Curvature on Transmission Conditions , 2014 .

[28]  Soheil Hajian.  An Optimized Schwarz Algorithm for a Discontinuous Galerkin Method , 2016 .

[29]  Martin J. Gander,et al.  Analysis of Schwarz Methods for a Hybridizable Discontinuous Galerkin Discretization , 2014, SIAM J. Numer. Anal..

[30]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[31]  Martin J. Gander,et al.  Optimized Schwarz Methods for Circular Domain Decompositions with Overlap , 2014, SIAM J. Numer. Anal..