Analysis and optimization of the generalized Schwarz method for elliptic problems with application to fluid–structure interaction

We propose a unified convergence analysis of the generalized Schwarz method applied to a linear elliptic problem for a general interface (flat, cylindrical or spherical) in any dimension. In particular, we provide the exact convergence set of the interface symbols related to the operators involved in the transmission conditions. We also provide a general procedure to obtain estimates of the optimized interface symbols within the constants. We apply such general results to a simple fluid–structure interaction model problem given by the interaction between an incompressible, inviscid fluid and the wave equation. Finally, we assess the effectiveness of the theoretical findings through three-dimensional numerical experiments in the haemodynamic context, obtained by solving the coupling between the Navier–Stokes equations and the linear infinitesimal elasticity.

[1]  F. Nobile,et al.  Optimization of Robin-Robin partitioned procedures in Fluid-Structure Interaction Problems , 2010 .

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

[3]  Fabio Nobile,et al.  Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems , 2009 .

[4]  Christian Vergara,et al.  Optimized Schwarz Methods for the Diffusion-Reaction Problem with Cylindrical Interfaces , 2013, SIAM J. Numer. Anal..

[5]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[6]  Martin J. Gander,et al.  shallow-water equations: preliminary results , 2022 .

[7]  Fabio Nobile,et al.  Added-mass effect in the design of partitioned algorithms for fluid-structure problems , 2005 .

[8]  A. Quarteroni,et al.  Fluid–structure algorithms based on Steklov–Poincaré operators , 2006 .

[9]  Miguel A. Fernández,et al.  Robin Based Semi-Implicit Coupling in Fluid-Structure Interaction: Stability Analysis and Numerics , 2009, SIAM J. Sci. Comput..

[10]  Fabio Nobile,et al.  Inexact accurate partitioned algorithms for fluid-structure interaction problems with finite elasticity in haemodynamics , 2014, J. Comput. Phys..

[11]  Miguel A. Fernández,et al.  A projection semi‐implicit scheme for the coupling of an elastic structure with an incompressible fluid , 2007 .

[12]  Alfio Quarteroni,et al.  Cardiovascular mathematics : modeling and simulation of the circulatory system , 2009 .

[13]  Annalisa Quaini,et al.  Splitting Methods Based on Algebraic Factorization for Fluid-Structure Interaction , 2008, SIAM J. Sci. Comput..

[14]  Frédéric Nataf,et al.  The optimized order 2 method: application to convection-diffusion problems , 1999 .

[15]  E. Ramm,et al.  Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows , 2007 .

[16]  F. Magoulès,et al.  Non-overlapping Schwarz methods with optimized transmission conditions for the H , 2004 .

[17]  W. Wall,et al.  Fixed-point fluid–structure interaction solvers with dynamic relaxation , 2008 .

[18]  J-F Gerbeau,et al.  External tissue support and fluid–structure simulation in blood flows , 2012, Biomechanics and modeling in mechanobiology.

[19]  Fabio Nobile,et al.  Partitioned Algorithms for Fluid-Structure Interaction Problems in Haemodynamics , 2012, Milan Journal of Mathematics.

[20]  Olivier Dubois,et al.  Optimized Schwarz Methods with Robin Conditions for the Advection-Diffusion Equation , 2007 .

[21]  Luca Gerardo-Giorda,et al.  Analysis and Optimization of Robin-Robin Partitioned Procedures in Fluid-Structure Interaction Problems , 2010, SIAM J. Numer. Anal..

[22]  O. Widlund,et al.  Domain Decomposition Methods in Science and Engineering XVI , 2007 .

[23]  Fabio Nobile,et al.  Time accurate partitioned algorithms for the solution of fluid–structure interaction problems in haemodynamics , 2013 .

[24]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[25]  G. Folland.  Introduction to Partial Differential Equations , 1976 .

[26]  J. Halleux,et al.  An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions , 1982 .

[27]  Alfio Quarteroni,et al.  On the physical consistency between three-dimensional and one-dimensional models in haemodynamics , 2013, J. Comput. Phys..

[28]  Martin J. Gander,et al.  Optimized Schwarz Methods for Maxwell's Equations , 2006, SIAM J. Sci. Comput..

[29]  Martin J. Gander,et al.  Optimized Schwarz Methods without Overlap for the Helmholtz Equation , 2002, SIAM J. Sci. Comput..

[30]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[31]  F. NOBILE,et al.  An Effective Fluid-Structure Interaction Formulation for Vascular Dynamics by Generalized Robin Conditions , 2008, SIAM J. Sci. Comput..

[32]  Bruno Stupfel,et al.  Improved transmission conditions for a one-dimensional domain decomposition method applied to the solution of the Helmholtz equation , 2010, J. Comput. Phys..

[33]  Fabio Nobile,et al.  Fluid-structure partitioned procedures based on Robin transmission conditions , 2008, J. Comput. Phys..