Universität zu Köln

Heinlein, Alexander ORCID: 0000-0003-1578-8104 and Lanser, Martin (2019). Additive and Hybrid Nonlinear Two-Level Schwarz Methods and Energy Minimizing Coarse Spaces for Unstructured Grids. Technical Report.

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Abstract

Nonlinear domain decomposition (DD) methods, such as, e.g., ASPIN (Additive Schwarz Preconditioned Inexact Newton), RASPEN (Restricted Additive Schwarz Preconditioned Inexact Newton), Nonlinear-FETI-DP, or Nonlinear-BDDC methods, can be reasonable alternatives to classical Newton-Krylov-DD methods for the solution of sparse nonlinear systems of equations, e.g., arising from a discretization of a nonlinear partial differential equation. These nonlinear DD approaches are often able to effectively tackle unevenly distributed nonlinearities and outperform Newton’s method with respect to convergence speed as well as global convergence behavior. Furthermore, they often improve parallel scalability due to a superior ratio of local to global work. Nonetheless, as for linear DD methods, it is often necessary to incorporate an appropriate coarse space in a second level to obtain numerical scalability for increasing numbers of subdomains. In addition to that, an appropriate coarse space can also improve the nonlinear convergence of nonlinear DD methods. In this paper, four variants how to integrate coarse spaces in nonlinear Schwarz methods in an additive or multiplicative way using Galerkin projections are introduced. These new variants can be interpreted as natural nonlinear equivalents to well-known linear additive and hybrid two-level Schwarz preconditioners. Furthermore, they facilitate the use of various coarse spaces, e.g., coarse spaces based on energy-minimizing extensions, which can easily be used for irregular domain decompositions, as, e.g., obtained by graph partitioners. In particular, Multiscale Finite Element Method (MsFEM) type coarse spaces are considered, and it is shown that they outperform classical approaches for certain heterogeneous nonlinear problems. The new approaches are then compared with classical Newton-Krylov-DD and nonlinear one-level Schwarz approaches for different homogeneous and heterogeneous model problems based on the p-Laplace operator.

Item Type:	Preprints, Working Papers or Reports (Technical Report)
Creators:	Creators Email ORCID ORCID Put Code Heinlein, Alexander alexander.heinlein@uni-koeln.de orcid.org/0000-0003-1578-8104 UNSPECIFIED Lanser, Martin martin.lanser@uni-koeln.de UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-98457
Series Name at the University of Cologne:	Technical report series. Center for Data and Simulation Science
Volume:	2019,17
Date:	23 July 2019
Language:	English
Faculty:	Central Institutions / Interdisciplinary Research Centers
Divisions:	Weitere Institute, Arbeits- und Forschungsgruppen > Center for Data and Simulation Science (CDS)
Subjects:	Natural sciences and mathematics Mathematics Technology (Applied sciences)
Uncontrolled Keywords:	Keywords Language nonlinear preconditioning English inexact Newton methods English nonlinear Schwarz methods English nonlinear domain decomposition English multiscale coarse spaces English ASPIN English RASPEN English
URI:	http://kups.ub.uni-koeln.de/id/eprint/9845