11 Least-squares approaches for the 2D Navier–Stokes system
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Jérôme Lemoine
Abstract
We analyze a least-squares approach to approximate weak solutions of the 2D Navier-Stokes system. In the first part, we consider a steady case and introduce a quadratic functional based on a weak norm of the state equation. We construct a minimizing sequence for the functional that strongly converges to a solution of the equation. After a finite number of iterates related to the value of the viscosity constant, the convergence is quadratic, from any initial guess. We then apply iteratively the analysis on the backward Euler scheme associated with the unsteady Navier-Stokes equation and prove the convergence of the iterative process uniformly with respect to the time discretization. In a second part, we reproduce the analysis for the unsteady case by introducing a space-time least-squares functional. This allows us to alleviate the smallness property on the data, assumed the steady case. The method turns out to be related to the globally convergent damped Newton approach applied to the Navier- Stokes operator, in contrast to the standard Newton method used to solve the weak formulation of the Navier-Stokes system. Numerical experiments illustrates our analysis.
Abstract
We analyze a least-squares approach to approximate weak solutions of the 2D Navier-Stokes system. In the first part, we consider a steady case and introduce a quadratic functional based on a weak norm of the state equation. We construct a minimizing sequence for the functional that strongly converges to a solution of the equation. After a finite number of iterates related to the value of the viscosity constant, the convergence is quadratic, from any initial guess. We then apply iteratively the analysis on the backward Euler scheme associated with the unsteady Navier-Stokes equation and prove the convergence of the iterative process uniformly with respect to the time discretization. In a second part, we reproduce the analysis for the unsteady case by introducing a space-time least-squares functional. This allows us to alleviate the smallness property on the data, assumed the steady case. The method turns out to be related to the globally convergent damped Newton approach applied to the Navier- Stokes operator, in contrast to the standard Newton method used to solve the weak formulation of the Navier-Stokes system. Numerical experiments illustrates our analysis.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1 Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE 1
- 2 Pointwise moving control for the 1-D wave equation 33
- 3 Limits of stabilizability for a semilinear model for gas pipeline flow 59
- 4 Minimal cost-time strategies for mosquito population replacement 73
- 5 The sterile insect technique used as a barrier control against reinfestation 91
- 6 Variational discretization approach applied to an optimal control problem with bounded measure controls 113
- 7 An optimal control problem for equations with p-structure and its finite element discretization 137
- 8 Unstructured space-time finite element methods for optimal sparse control of parabolic equations 167
- 9 An adaptive finite element approach for lifted branched transport problems 189
- 10 High-order homogenization of the Poisson equation in a perforated periodic domain 237
- 11 Least-squares approaches for the 2D Navier–Stokes system 285
- 12 Numerical issues and turnpike phenomenon in optimal shape design 343
- 13 Feedback stabilization of Cahn–Hilliard phase-field systems 367
- 14 Ensemble Kalman filter for neural network-based one-shot inversion 393
- 15 Deep learning in high dimension: ReLU neural network expression for Bayesian PDE inversion 419
- Index 463
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1 Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE 1
- 2 Pointwise moving control for the 1-D wave equation 33
- 3 Limits of stabilizability for a semilinear model for gas pipeline flow 59
- 4 Minimal cost-time strategies for mosquito population replacement 73
- 5 The sterile insect technique used as a barrier control against reinfestation 91
- 6 Variational discretization approach applied to an optimal control problem with bounded measure controls 113
- 7 An optimal control problem for equations with p-structure and its finite element discretization 137
- 8 Unstructured space-time finite element methods for optimal sparse control of parabolic equations 167
- 9 An adaptive finite element approach for lifted branched transport problems 189
- 10 High-order homogenization of the Poisson equation in a perforated periodic domain 237
- 11 Least-squares approaches for the 2D Navier–Stokes system 285
- 12 Numerical issues and turnpike phenomenon in optimal shape design 343
- 13 Feedback stabilization of Cahn–Hilliard phase-field systems 367
- 14 Ensemble Kalman filter for neural network-based one-shot inversion 393
- 15 Deep learning in high dimension: ReLU neural network expression for Bayesian PDE inversion 419
- Index 463
