In this work, we focus on the Optimized Schwarz Method for circular flat interfaces and geometric heterogeneous coupling arising when cylindrical geometries are coupled along the axial direction. In the first case, we provide a convergence analysis for the diffusion-reaction problem and jumping coefficients and we apply the general optimization procedure developed in Gigante and Vergara (Numer. Math. 131 (2015) 369–404). In the numerical simulations, we discuss how to choose the range of frequencies in the optimization and the influence of the Finite Element and projection errors on the convergence. In the second case, we consider the coupling between a three-dimensional and a one-dimensional diffusion-reaction problem and we develop a new optimization procedure. The numerical results highlight the suitability of the theoretical findings.
Mots clés : Optimized Schwarz Method, cylindrical domains, geometric multiscale, Bessel functions
@article{M2AN_2018__52_4_1597_0,
author = {Gigante, Giacomo and Vergara, Christian},
title = {Optimized {Schwarz} methods for the coupling of cylindrical geometries along the axial direction},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1597--1615},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2018039},
zbl = {1407.65260},
mrnumber = {3878606},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018039/}
}
TY - JOUR AU - Gigante, Giacomo AU - Vergara, Christian TI - Optimized Schwarz methods for the coupling of cylindrical geometries along the axial direction JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1597 EP - 1615 VL - 52 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018039/ DO - 10.1051/m2an/2018039 LA - en ID - M2AN_2018__52_4_1597_0 ER -
%0 Journal Article %A Gigante, Giacomo %A Vergara, Christian %T Optimized Schwarz methods for the coupling of cylindrical geometries along the axial direction %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1597-1615 %V 52 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018039/ %R 10.1051/m2an/2018039 %G en %F M2AN_2018__52_4_1597_0
Gigante, Giacomo; Vergara, Christian. Optimized Schwarz methods for the coupling of cylindrical geometries along the axial direction. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1597-1615. doi : 10.1051/m2an/2018039. http://www.numdam.org/articles/10.1051/m2an/2018039/
[1] , and , Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227 (2008) 7027–7051. | DOI | MR | Zbl
[2] , and , Modeling dimensionally-heterogeneous problems: analysis, approximation and applications. Numer. Math. 119 (2011) 299–335. | DOI | MR | Zbl
[3] , and , On the continuity of mean total normal stress in geometrical multiscale cardiovascular problems. J. Comput. Phys. 51 (2013) 136–155. | DOI | MR | Zbl
[4] , and , A unified variational approach for coupling 3d–1d models and its blood flow applications. Comput. Methods Appl. Mech. Eng. 196 (2007) 4391–4410. | DOI | MR | Zbl
[5] , and , Optimized Schwarz Methods for Maxwell’s equations. SIAM J. Sci. Comput. 31 (2009) 2193–2213. | DOI | MR | Zbl
[6] and , Numerical approximation of a quasi-newtonian Stokes flow problem with defective boundary conditions. SIAM J. Numer. Anal. 45 (2007) 2120–2140. | DOI | MR | Zbl
[7] , , and , On the coupling of 3D an 1D Navier–Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191 (2001) 561–582. | DOI | MR | Zbl
[8] , , and , Numerical treatment of defective boundary conditions for the Navier–Stokes equation. SIAM J. Numer. Anal. 40 (2002) 376–401. | DOI | MR | Zbl
[9] , and , On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations. ESAIM: M2AN 41 (2007) 743–769. | DOI | Numdam | MR | Zbl
[10] , and , On the physical consistency between three-dimensional and one-dimensional models in haemodynamics. J. Comput. Phys. 244 (2013) 97–112. | DOI | MR | Zbl
[11] , and , A new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics. SIAM J. Numer. Anal. 46 (2008) 2769–2794. | DOI | MR | Zbl
[12] , and , Flow rate boundary problems for an incompressible fluid in deformable domains: formulations and solution methods. Comput. Methods Appl. Mech. Eng. 199 (2009) 677–688. | DOI | MR | Zbl
[13] and , Prescription of general defective boundary conditions in fluid-dynamics. Milan J. Math. 80 (2012) 333–350. | DOI | MR | Zbl
[14] and , Analysis and approximation of the cross model for quasi-newtonian flows with defective boundary conditions. Appl. Math. Comp. 222 (2013) 244–254. | DOI | MR | Zbl
[15] , and , Approximation of viscoelastic flows with defective boundary conditions. J. Non Newt. Fl. Mech. 169–170 (2012) 104–113. | DOI
[16] , Optimized Schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699–731. | DOI | MR | Zbl
[17] , and , Optimized Schwarz methods without overlap for the Helmholtz equation. SIAM J. Sci. Comput. 24 (2002) 38–60. | DOI | MR | Zbl
[18] and , Optimized Schwarz methods for circular domain decompositions with overlap. SIAM J. Numer. Anal. 52 (2014) 1981–2004. | DOI | MR | Zbl
[19] and , Optimized Schwarz Methods with nonoverlapping circular domain decomposition. Math. Comp. 86 (2017) 637–660. | DOI | MR | Zbl
[20] L. Gerardo Giorda, and , Analysis and optimization of Robin–Robin partitioned procedures in fluid-structure interaction problems. SIAM J. Numer. Anal. 48 (2010) 2091–2116. | DOI | MR | Zbl
[21] , and , Optimized Schwarz methods for the diffusion-reaction problem with cylindrical interfaces. SIAM J. Numer. Anal. 51 (2013) 3402–3420. | DOI | MR | Zbl
[22] and , Analysis and optimization of the generalized Schwarz method for elliptic problems with application to fluid-structure interaction. Numer. Math. 131 (2015) 369–404. | DOI | MR | Zbl
[23] and , Optimized Schwarz method for the fluid-structure interaction with cylindrical interfaces, edited by , , , and . In: Proc. of the XXII International Conference on Domain Decomposition Methods. Springer Nature Switzerland (2016) 521–529. | DOI | MR
[24] , and , Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. Int. J. Num. Methods Fluids 22 (1996) 325–352. | DOI | MR | Zbl
[25] , Optimized Krylov–Ventcell method. Application to convection-diffusion problems, edited by , and . In: Proc. of the Ninth International Conference on Domain Decomposition Methods. Springer, Berlin (1998) 382–389.
[26] , and , The optimized order 2 method. Application to convection-diffusion problems. Fut. Gen. Comput. Syst. 18 (2001) 17–30. | DOI | Zbl
[27] , Special Functions and Their Applications. Courier Dover Publications, Mineola, NY, USA (1972). | MR | Zbl
[28] , Optimal control for quasi-newtonian flows with defective boundary conditions. Comput. Methods Appl. Mech. Eng. 200 (2011) 2498–2506. | DOI | MR | Zbl
[29] , and , Partitioned analysis for dimensionally-heterogeneous hydraulic networks. Mult. Model. Simul. 9 (2011) 872–903. | DOI | MR | Zbl
[30] , On the Schwarz alternating method III, edited by , , and , In: Proc. of the Third International Symposium on Domain Decomposition Methods for PDE’s. Siam, Philadelphia (1990) 202–223. | MR | Zbl
[31] F. Magoulès, and , Non-overlapping Schwarz method with optimized transmission conditions for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 193 (2004) 4797–4818. | DOI | MR | Zbl
[32] , , , and , Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels. Multiscale Model Simul. 11 (2013) 474–506. | DOI | MR | Zbl
[33] , Coupling 3d and 1d fluid–structure-interaction models for wave propagation in flexible vessels using a finite volume pressure-correction scheme. Comm. Numer. Meth. Eng. 25 (2009) 533–551. | DOI | MR | Zbl
[34] , , and , Numerical treatment of boundary conditions to replace lateral branches in haemodynamics. Int. J. Numer. Meth. Biomed. Eng. 28 (2012) 1165–1183. | DOI | MR
[35] , , , J Cote and , Optimized Schwarz methods with an overset grid for the shallow-water equations: preliminary results. Appl. Num. Math. 58 (2008) 459–471. | DOI | MR | Zbl
[36] , and , The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications. Acta Numer. 26 (2017) 365–590. | DOI | MR | Zbl
[37] , and , Geometric multiscale modeling of the cardiovascular system, between theory and practice. Comput. Methods Appl. Mech. Eng. 302 (2016) 193–252. | DOI | MR | Zbl
[38] , Improved transmission conditions for a one-dimensional domain decomposition method applied to the solution of the Helmhotz equation. J. Comput. Phys. 229 (2010) 851–874. | DOI | MR | Zbl
[39] and , Flow rate defective boundary conditions in haemodinamics simulations. Int. J. Num. Methods Fluids 47 (2005) 803–816. | DOI | MR | Zbl
[40] and , An approximate method for solving incompressible Navier-Stokes problems with flow rate conditions. Comput. Methods Appl. Mech. Eng. 196 (2007) 1685–1700. | DOI | MR | Zbl
[41] , Nitsche’s method for defective boundary value problems in incompressibile fluid-dynamics. J. Sci. Comput. 46 (2011) 100–123. | DOI | MR | Zbl
[42] I.E. Vignon-Clementel, , and , Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure waves in arteries. Comput. Methods Appl. Mech. Eng. 195 (2006) 3776–3996. | DOI | MR | Zbl
[43] , and , Generalized fictitious methods for fluid-structure interactions: analysis and simulations. J. Comput. Phys. 245 (2013) 317–346. | DOI | MR | Zbl
[44] , Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty technique. Comput. Methods Appl. Mech. Eng. 198 (2009) 3026–3038. | DOI | MR | Zbl
Cité par Sources :
