Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi’s and Babuška’s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi’s saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuška’s inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.

[1]  R. Sani,et al.  Incompressible Flow and the Finite Element Method, Volume 1, Advection-Diffusion and Isothermal Laminar Flow , 1998 .

[2]  Sidney Yip,et al.  Handbook of Materials Modeling , 2005 .

[3]  Max Gunzburger,et al.  3 – Finite Element Spaces , 1989 .

[4]  Gianluigi Rozza,et al.  REDUCED BASIS METHODS AND A POSTERIORI ERROR ESTIMATORS FOR HEAT TRANSFER PROBLEMS , 2009 .

[5]  Yvon Maday,et al.  A reduced basis element method for the steady stokes problem , 2006 .

[6]  Gianluigi Rozza,et al.  On optimization, control and shape design of an arterial bypass , 2005 .

[7]  Anthony T. Patera,et al.  A natural-norm Successive Constraint Method for inf-sup lower bounds , 2010 .

[8]  Gianluigi Rozza,et al.  Real time reduced basis techniques for arterial bypass geometries , 2005 .

[9]  J. Rappaz,et al.  Numerical analysis for nonlinear and bifurcation problems , 1997 .

[10]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .

[11]  Gianluigi Rozza,et al.  Reduced Basis Approximation for Shape Optimization in Thermal Flows with a Parametrized Polynomial Geometric Map , 2010 .

[12]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[13]  Einar M. Rønquist,et al.  Spectral and high order methods for partial differential equations : selected papers from the ICOSAHOM '09 conference, June 22-26, Trondheim, Norway , 2010 .

[14]  I. Babuska.  Error-bounds for finite element method , 1971 .

[15]  Ahmed K. Noor,et al.  On making large nonlinear problems small , 1982 .

[16]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[17]  Franco Brezzi Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods (Springer Series in Computational Mathematics) , 1991 .

[18]  D. Rovas,et al.  Reduced--Basis Output Bound Methods for Parametrized Partial Differential Equations , 2002 .

[19]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[20]  Gianluigi Rozza,et al.  Reduced basis approximation and error bounds for potential flows in parametrized geometries , 2011 .

[21]  A. Quarteroni.  Numerical Models for Differential Problems , 2009 .

[22]  Max Gunzburger,et al.  Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms , 1989 .

[23]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .

[24]  G. Rozza,et al.  Model Order Reduction by geometrical parametrization for shape optimization in computational fluid dynamics , 2010 .

[25]  A. Quarteroni,et al.  Numerical solution of parametrized Navier–Stokes equations by reduced basis methods , 2007 .

[26]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[27]  Nguyen Ngoc Cuong,et al.  Certified Real-Time Solution of Parametrized Partial Differential Equations , 2005 .

[28]  Gianluigi Rozza.  Shape design by optimal flow control and reduced basis techniques , 2005 .

[29]  J. Szmelter.  Incompressible flow and the finite element method , 2001 .

[30]  Gianluigi Rozza,et al.  Reduced basis methods for Stokes equations in domains with non-affine parameter dependence , 2009 .

[31]  Janet S. Peterson,et al.  The Reduced Basis Method for Incompressible Viscous Flow Calculations , 1989 .

[32]  Ludmil T. Zikatanov,et al.  Some observations on Babu\vs}ka and Brezzi theories , 2003, Numerische Mathematik.

[33]  Barbara I. Wohlmuth,et al.  A Reduced Basis Method for Parametrized Variational Inequalities , 2012, SIAM J. Numer. Anal..

[34]  G. Rozza,et al.  Parametric free-form shape design with PDE models and reduced basis method , 2010 .

[35]  Gianluigi Rozza,et al.  Real-time reduced basis techniques for Navier-Stokes equations: Optimization of parametrized bypass configurations , 2006 .

[36]  Gianluigi Rozza,et al.  Reduced basis method for linear elasticity problems with many parameters , 2008 .

[37]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[38]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .

[39]  Luca Dedè,et al.  Optimal flow control for Navier–Stokes equations: drag minimization , 2007 .

[40]  Gianluigi Rozza,et al.  Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity , 2009, J. Comput. Phys..

[41]  N. Nguyen,et al.  REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS , 2011 .

[42]  Simone Deparis,et al.  Reduced Basis Error Bound Computation of Parameter-Dependent Navier-Stokes Equations by the Natural Norm Approach , 2008, SIAM J. Numer. Anal..

[43]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[44]  Karen Veroy,et al.  REDUCED BASIS A POSTERIORI ERROR BOUNDS FOR THE STOKES EQUATIONS IN PARAMETRIZED DOMAINS: A PENALTY APPROACH , 2010 .

[45]  Simone Deparis,et al.  Stabilized Reduced Basis Approximation of Incompressible Three-Dimensional Navier-Stokes Equations in Parametrized Deformed Domains , 2012, J. Sci. Comput..

[46]  A. Patera,et al.  A Successive Constraint Linear Optimization Method for Lower Bounds of Parametric Coercivity and Inf-Sup Stability Constants , 2007 .

[47]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[48]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[49]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[50]  Luca Dedè,et al.  Adaptive and Reduced Basis methods for optimal control problems in environmental applications , 2008 .

[51]  A. Quarteroni,et al.  Shape optimization for viscous flows by reduced basis methods and free‐form deformation , 2012 .

[52]  G. Rozza,et al.  On the stability of the reduced basis method for Stokes equations in parametrized domains , 2007 .