Convergence of linear and nonlinear Neumann–Neumann method for the Cahn–Hilliard equation

In this paper, we propose and analyze a non-overlapping substructuring type algorithm for the Cahn–Hilliard equation. Being a nonlinear equation, it is of great importance to develop a robust numerical method for investigating the solution behaviour of the CH equation. We present the formulation of...

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Published in:	Japan journal of industrial and applied mathematics Vol. 41; no. 1; pp. 211 - 232
Main Author:	Garai, Gobinda
Format:	Journal Article
Language:	English
Published:	Tokyo Springer Japan 01.01.2024 Springer Nature B.V
Subjects:	Algorithms Applications of Mathematics Approximation Boundary conditions Computational Mathematics and Numerical Analysis Convergence Decomposition Energy Mathematics Mathematics and Statistics Nonlinear equations Nonlinearity Numerical methods Original Paper 65M15 Neumann–Neumann Parallel computing 65M55 Domain decomposition 65Y05 Cahn–Hilliard equation
ISSN:	0916-7005, 1868-937X
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Abstract	In this paper, we propose and analyze a non-overlapping substructuring type algorithm for the Cahn–Hilliard equation. Being a nonlinear equation, it is of great importance to develop a robust numerical method for investigating the solution behaviour of the CH equation. We present the formulation of the Neumann–Neumann method applied to the CH equation and study its convergence behaviour in one and two spatial dimension for two subdomains and also extend the method for logarithmic nonlinearity. We also present the nonlinear NN method for the CH equation. We illustrate the theoretical results by providing numerical examples.
AbstractList	In this paper, we propose and analyze a non-overlapping substructuring type algorithm for the Cahn–Hilliard equation. Being a nonlinear equation, it is of great importance to develop a robust numerical method for investigating the solution behaviour of the CH equation. We present the formulation of the Neumann–Neumann method applied to the CH equation and study its convergence behaviour in one and two spatial dimension for two subdomains and also extend the method for logarithmic nonlinearity. We also present the nonlinear NN method for the CH equation. We illustrate the theoretical results by providing numerical examples.
Author	Garai, Gobinda
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Cites_doi	10.1137/20M1316317 10.1016/0362-546X(94)00205-V 10.1016/0167-2789(84)90180-5 10.1007/978-1-4612-4248-2_9 10.1109/TIP.2006.887728 10.1063/1.1744102 10.1016/j.cnsns.2023.107175 10.1016/0001-6160(61)90182-1 10.1016/j.commatsci.2013.08.027 10.1557/PROC-529-39 10.1007/978-3-319-52389-7_11 10.1007/s10013-018-0316-9 10.1007/978-1-4419-7288-0 10.1002/cpa.10020 10.1007/b137868 10.1007/978-3-030-56750-7_8 10.1137/0723075 10.1016/0377-0427(91)90150-I 10.1090/S0025-5718-96-00757-0 10.1080/03605308908820597
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References	BertozziALEsedoḡluSGilletteAInpainting of binary images using the Cahn–Hilliard equationIEEE Trans. Image Process.200716285291246016710.1109/TIP.2006.887728 Chaouqui, F., Gander, M. J., Santugini-Repiquet, K.: On nilpotent subdomain iterations. In: Domain Decomposition Methods in Science and Engineering XXIII. Springer, Berlin, pp. 125–133 (2017) BjørstadPEWidlundOBIterative methods for the solution of elliptic problems on regions partitioned into substructuresSIAM J. Numer. Anal.1986231097112086594510.1137/0723075 Novick-CohenASegelLANonlinear aspects of the Cahn–Hilliard equationPhys. D19841027729876347310.1016/0167-2789(84)90180-5 ToselliAWidlundOBDomain Decomposition Methods, Algorithms and Theory2005BerlinSpringer10.1007/b137868 MandelJBrezinaMBalancing domain decomposition for problems with large jumps in coefficientsMath. Comput.19966513871401135120410.1090/S0025-5718-96-00757-0 Le TallecPDe RoeckY-HVidrascuMDomain decomposition methods for large linearly elliptic three-dimensional problemsJ. Comput. Appl. Math.19913493117109519810.1016/0377-0427(91)90150-I PavarinoLFWidlundOBBalancing Neumann–Neumann methods for incompressible stokes equationsCommun. Pure Appl. Math. J. Iss. Courant Inst. Math. Sci.200255302335186636610.1002/cpa.10020 GaraiGMandalBCConvergence of substructuring methods for the Cahn–Hilliard equationCommun. Nonlinear Sci. Numer. Simul.2023120455230010.1016/j.cnsns.2023.107175 NicolaenkoBScheurerBTemamRSome global dynamical properties of a class of pattern formation equationsComm. Partial Differ. Equ.19891424529797697310.1080/03605308908820597 CahnJWHilliardWFree energy of a nonuniform system. i. interfacial free energyJ. Chem. Phys.19582825826710.1063/1.1744102 BakJNewmanDJNewmanDJComplex Analysis2010BerlinSpringer10.1007/978-1-4419-7288-0 CarlenzoliCQuarteroniAAdaptive domain decomposition methods for advection-diffusion problemsModeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations1995BerlinSpringer16518610.1007/978-1-4612-4248-2_9 Eyre, D.J., Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Computational and Mathematical Models of Microstructural Evolution, San Francisco, CA,: vol. 529 of Mater. Res. Soc. Sympos. Proc. MRS, Warrendale, PA vol. 1998, pp. 39–46 (1998) DebusscheADettoriLOn the Cahn–Hilliard equation with a logarithmic free energyNonlinear Anal. Theory Methods Appl.19952414911514132793010.1016/0362-546X(94)00205-V LionsP-LOn the Schwarz alternating method IDomain Decomposition Methods for Partial Differential Equations (Paris, 1987)1988PhiladelphiaSIAM142 LeeSLeeCLeeHGKimJComparison of different numerical schemes for the Cahn–Hilliard equationJ. KSIAM2013171972073105844 ChaouquiFGanderMJSantugini-RepiquetKA continuous analysis of Neumann–Neumann methods: scalability and new coarse spacesSIAM J. Sci. Comput.202042A3785A3811418109910.1137/20M1316317 Chaouqui, F., Gander, M. J., Santugini-Repiquet, K.: A local coarse space correction leading to a well-posed continuous Neumann–Neumann method in the presence of cross points. In: International Conference on Domain Decomposition Methods. Springer, Berlin, pp. 83–91 (2018) NicolaenkoBScheurerBLow-dimensional behavior of the pattern formation Cahn–Hilliard equationNorth-Holland Mathematics Studies1985OxfordElsevier323336 CahnJWOn spinodal decompositionActa Metall.1961979580110.1016/0001-6160(61)90182-1 Eyre, D.J.: An unconditionally stable one-step scheme for gradient systems, Unpublished article, (1998) ChaouquiFCiaramellaGGanderMJVanzanTOn the scalability of classical one-level domain-decomposition methodsVietnam J. Math.20184610531088387879310.1007/s10013-018-0316-9 BourgatJ-FGlowinskiRLe TallecPVidrascuMVariational Formulation and Algorithm for Trace Operation in Domain Decomposition Calculations, Domain Decomposition Methods for Partial Differential Equations1989PhiladelphiaSIAM316 LeeDHuhJ-YJeongDShinJYunAKimJPhysical, mathematical, and numerical derivations of the Cahn–Hilliard equationComput. Mater. Sci.20148121622510.1016/j.commatsci.2013.08.027 J-F Bourgat (600_CR4) 1989 P Le Tallec (600_CR16) 1991; 34 B Nicolaenko (600_CR21) 1985 J Mandel (600_CR20) 1996; 65 B Nicolaenko (600_CR22) 1989; 14 C Carlenzoli (600_CR7) 1995 PE Bjørstad (600_CR3) 1986; 23 S Lee (600_CR17) 2013; 17 JW Cahn (600_CR6) 1958; 28 600_CR14 600_CR13 AL Bertozzi (600_CR2) 2007; 16 JW Cahn (600_CR5) 1961; 9 600_CR9 600_CR8 A Novick-Cohen (600_CR23) 1984; 10 LF Pavarino (600_CR24) 2002; 55 A Debussche (600_CR12) 1995; 24 J Bak (600_CR1) 2010 F Chaouqui (600_CR11) 2020; 42 P-L Lions (600_CR19) 1988 F Chaouqui (600_CR10) 2018; 46 G Garai (600_CR15) 2023; 120 D Lee (600_CR18) 2014; 81 A Toselli (600_CR25) 2005
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Math.20184610531088387879310.1007/s10013-018-0316-9 – reference: ToselliAWidlundOBDomain Decomposition Methods, Algorithms and Theory2005BerlinSpringer10.1007/b137868 – reference: Le TallecPDe RoeckY-HVidrascuMDomain decomposition methods for large linearly elliptic three-dimensional problemsJ. Comput. Appl. Math.19913493117109519810.1016/0377-0427(91)90150-I – reference: BakJNewmanDJNewmanDJComplex Analysis2010BerlinSpringer10.1007/978-1-4419-7288-0 – reference: ChaouquiFGanderMJSantugini-RepiquetKA continuous analysis of Neumann–Neumann methods: scalability and new coarse spacesSIAM J. Sci. Comput.202042A3785A3811418109910.1137/20M1316317 – reference: PavarinoLFWidlundOBBalancing Neumann–Neumann methods for incompressible stokes equationsCommun. Pure Appl. Math. J. Iss. Courant Inst. Math. Sci.200255302335186636610.1002/cpa.10020 – reference: Eyre, D.J.: An unconditionally stable one-step scheme for gradient systems, Unpublished article, (1998) – reference: Chaouqui, F., Gander, M. J., Santugini-Repiquet, K.: A local coarse space correction leading to a well-posed continuous Neumann–Neumann method in the presence of cross points. In: International Conference on Domain Decomposition Methods. Springer, Berlin, pp. 83–91 (2018) – reference: DebusscheADettoriLOn the Cahn–Hilliard equation with a logarithmic free energyNonlinear Anal. Theory Methods Appl.19952414911514132793010.1016/0362-546X(94)00205-V – reference: Eyre, D.J., Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Computational and Mathematical Models of Microstructural Evolution, San Francisco, CA,: vol. 529 of Mater. Res. Soc. Sympos. Proc. MRS, Warrendale, PA vol. 1998, pp. 39–46 (1998) – reference: Novick-CohenASegelLANonlinear aspects of the Cahn–Hilliard equationPhys. D19841027729876347310.1016/0167-2789(84)90180-5 – reference: LeeDHuhJ-YJeongDShinJYunAKimJPhysical, mathematical, and numerical derivations of the Cahn–Hilliard equationComput. Mater. Sci.20148121622510.1016/j.commatsci.2013.08.027 – reference: BjørstadPEWidlundOBIterative methods for the solution of elliptic problems on regions partitioned into substructuresSIAM J. Numer. Anal.1986231097112086594510.1137/0723075 – reference: NicolaenkoBScheurerBLow-dimensional behavior of the pattern formation Cahn–Hilliard equationNorth-Holland Mathematics Studies1985OxfordElsevier323336 – reference: Chaouqui, F., Gander, M. J., Santugini-Repiquet, K.: On nilpotent subdomain iterations. In: Domain Decomposition Methods in Science and Engineering XXIII. Springer, Berlin, pp. 125–133 (2017) – reference: GaraiGMandalBCConvergence of substructuring methods for the Cahn–Hilliard equationCommun. Nonlinear Sci. Numer. Simul.2023120455230010.1016/j.cnsns.2023.107175 – reference: CahnJWOn spinodal decompositionActa Metall.1961979580110.1016/0001-6160(61)90182-1 – reference: CarlenzoliCQuarteroniAAdaptive domain decomposition methods for advection-diffusion problemsModeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations1995BerlinSpringer16518610.1007/978-1-4612-4248-2_9 – reference: CahnJWHilliardWFree energy of a nonuniform system. i. interfacial free energyJ. Chem. Phys.19582825826710.1063/1.1744102 – reference: LeeSLeeCLeeHGKimJComparison of different numerical schemes for the Cahn–Hilliard equationJ. KSIAM2013171972073105844 – reference: LionsP-LOn the Schwarz alternating method IDomain Decomposition Methods for Partial Differential Equations (Paris, 1987)1988PhiladelphiaSIAM142 – volume: 42 start-page: A3785 year: 2020 ident: 600_CR11 publication-title: SIAM J. Sci. 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Phys. doi: 10.1063/1.1744102 – volume: 120 year: 2023 ident: 600_CR15 publication-title: Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2023.107175 – volume: 17 start-page: 197 year: 2013 ident: 600_CR17 publication-title: J. KSIAM – volume: 9 start-page: 795 year: 1961 ident: 600_CR5 publication-title: Acta Metall. doi: 10.1016/0001-6160(61)90182-1 – start-page: 323 volume-title: North-Holland Mathematics Studies year: 1985 ident: 600_CR21 – volume: 81 start-page: 216 year: 2014 ident: 600_CR18 publication-title: Comput. Mater. Sci. doi: 10.1016/j.commatsci.2013.08.027 – ident: 600_CR13 doi: 10.1557/PROC-529-39 – ident: 600_CR9 doi: 10.1007/978-3-319-52389-7_11 – start-page: 1 volume-title: Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) year: 1988 ident: 600_CR19 – volume: 46 start-page: 1053 year: 2018 ident: 600_CR10 publication-title: Vietnam J. Math. doi: 10.1007/s10013-018-0316-9 – volume-title: Complex Analysis year: 2010 ident: 600_CR1 doi: 10.1007/978-1-4419-7288-0 – volume: 55 start-page: 302 year: 2002 ident: 600_CR24 publication-title: Commun. Pure Appl. Math. J. Iss. Courant Inst. Math. Sci. doi: 10.1002/cpa.10020 – volume-title: Domain Decomposition Methods, Algorithms and Theory year: 2005 ident: 600_CR25 doi: 10.1007/b137868 – ident: 600_CR8 doi: 10.1007/978-3-030-56750-7_8 – volume: 23 start-page: 1097 year: 1986 ident: 600_CR3 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0723075 – volume: 34 start-page: 93 year: 1991 ident: 600_CR16 publication-title: J. Comput. Appl. Math. doi: 10.1016/0377-0427(91)90150-I – volume: 65 start-page: 1387 year: 1996 ident: 600_CR20 publication-title: Math. Comput. doi: 10.1090/S0025-5718-96-00757-0 – volume: 14 start-page: 245 year: 1989 ident: 600_CR22 publication-title: Comm. Partial Differ. Equ. doi: 10.1080/03605308908820597 – ident: 600_CR14
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Snippet	In this paper, we propose and analyze a non-overlapping substructuring type algorithm for the Cahn–Hilliard equation. Being a nonlinear equation, it is of...
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SubjectTerms	Algorithms Applications of Mathematics Approximation Boundary conditions Computational Mathematics and Numerical Analysis Convergence Decomposition Energy Mathematics Mathematics and Statistics Nonlinear equations Nonlinearity Numerical methods Original Paper
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Title	Convergence of linear and nonlinear Neumann–Neumann method for the Cahn–Hilliard equation
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