Open accessJournal Article

# Analysis of continuous h −1 least-squares methods for the steady navier-stokes system

02 Mar 2021-Applied Mathematics and Optimization (Springer US)-Vol. 83, Iss: 1, pp 461-488
Abstract: We analyse two H −1 least-squares methods for the steady Navier-Stokes system of incompressible viscous fluids. Precisely, we show the convergence of minimizing sequences for the least-squares functional toward solutions. Numerical experiments support our analysis.

Topics: Least squares (60%)
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Open accessJournal Article
Jérôme Lemoine1, Arnaud Münch1Institutions (1)
Abstract: This work analyzes a least-squares method in order to solve implicit time schemes associated to the 2D and 3D Navier–Stokes system, introduced in 1979 by Bristeau, Glowinksi, Periaux, Perrier and Pironneau. Implicit time schemes reduce the numerical resolution of the Navier–Stokes system to multiple resolutions of steady Navier–Stokes equations. We first construct a minimizing sequence (by a gradient type method) for the least-squares functional which converges strongly and quadratically toward a solution of a steady Navier–Stokes equation from any initial guess. The method turns out to be related to the globally convergent damped Newton approach applied to the Navier–Stokes operator. Then, we apply iteratively the analysis on the fully implicit Euler scheme and show the convergence of the method uniformly with respect to the time discretization. Numerical experiments for 2D examples support our analysis.

7 Citations

Open accessJournal Article
Jérôme Lemoine1, Arnaud Münch1Institutions (1)
Abstract: We introduce and analyze a space-time least-squares method associated with the unsteady Navier–Stokes system. Weak solution in the two dimensional case and regular solution in the three dimensional case are considered. From any initial guess, we construct a minimizing sequence for the least-squares functional which converges strongly to a solution of the Navier–Stokes system. After a finite number of iterations related to the value of the viscosity coefficient, the convergence is quadratic. Numerical experiments within the two dimensional case support our analysis. This globally convergent least-squares approach is related to the damped Newton method.

Topics: Regular solution (56%), Newton's method (55%), Weak solution (54%)

6 Citations

Open accessPosted Content
Jérôme Lemoine1, Arnaud Münch1Institutions (1)
Abstract: We introduce and analyze a space-time least-squares method associated to the unsteady Navier-Stokes system. Weak solution in the two dimensional case and regular solution in the three dimensional case are considered. From any initial guess, we construct a minimizing sequence for the least-squares functional which converges strongly to a solution of the Navier-Stokes system. After a finite number of iterates related to the value of the viscosity constant, the convergence is quadratic. Numerical experiments within the two dimensional case support our analysis. This globally convergent least-squares approach is related to the damped Newton method when used to solve the Navier-Stokes system through a variational formulation.

Topics: Regular solution (57%), Newton's method (56%), Weak solution (55%) ... show more

4 Citations

Open accessPosted Content
Arnaud Münch1, Emmanuel Trélat2Institutions (2)
Abstract: It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation ∂tty − ∂xxy + g(y) = f 1ω, with Dirichlet boundary conditions, is exactly controllable in H 1 0 (0, 1) ∩ L 2 (0, 1) with controls f ∈ L 2 ((0, 1) × (0, T)), for any T > 0 and any nonempty open subset ω of (0, 1), assuming that g ∈ C 1 (R) does not grow faster than β|x| ln 2 |x| at infinity for some β > 0 small enough. The proof, based on the Leray-Schauder fixed point theorem, is however not constructive. In this article, we design a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations. Assuming that g does not grow faster than β ln 2 |x| at infinity for some β > 0 small enough and that g is uniformly Holder continuous on R with exponent s ∈ [0, 1], we design a least-squares algorithm yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order 1 + s after a finite number of iterations.

Topics: Constructive proof (53%), , Fixed-point theorem (51%)

4 Citations

Open accessPosted Content
Abstract: The null distributed controllability of the semilinear heat equation $y_t-\Delta y + g(y)=f \,1_{\omega}$, assuming that $g$ satisfies the growth condition $g(s)/(\vert s\vert \log^{3/2}(1+\vert s\vert))\rightarrow 0$ as $\vert s\vert \rightarrow \infty$ and that $g^\prime\in L^\infty_{loc}(\mathbb{R})$ has been obtained by Fernandez-Cara and Zuazua in 2000. The proof based on a fixed point argument makes use of precise estimates of the observability constant for a linearized heat equation. It does not provide however an explicit construction of a null control. Assuming that $g^\prime\in W^{s,\infty}(\mathbb{R})$ for one $s\in (0,1]$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach, generalizes Newton type methods and guarantees the convergence whatever be the initial element of the sequence. In particular, after a finite number of iterations, the convergence is super linear with a rate equal to $1+s$. Numerical experiments in the one dimensional setting support our analysis.

Topics: Type (model theory) (64%), Null (mathematics) (55%)

3 Citations

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23 results found

Open accessJournal Article
Abstract: This is a short presentation of the freefem++ software. In Section 1, we recall most of the characteristics of the software, In Section 2, we recall how to to build the weak form of a partial differential equation (PDE) from the strong form. In the 3 last sections, we present different examples and tools to illustrated the power of the software. First we deal with mesh adaptation for problems in two and three dimension, second, we solve numerically a problem with phase change and natural convection, and the finally to show the possibilities for HPC we solve a Laplace equation by a Schwarz domain decomposition problem on parallel computer.

2,275 Citations

Journal Article
Abstract: Etude de nouvelles methodes de descente suivant le gradient pour la solution approchee du probleme de minimisation sans contrainte. Analyse de la convergence

2,203 Citations

Open access
William W. Hager1, Hongchao Zhang1Institutions (1)
01 Jan 2005-
Abstract: This paper reviews the development of dierent versions of nonlinear conjugate gradient methods, with special attention given to global convergence properties.

689 Citations

Journal Article
Abstract: The Barzilai and Borwein gradient method for the solution of large scale unconstrained minimization problems is considered. This method requires few storage locations and very inexpensive computations. Furthermore, it does not guarantee descent in the objective function and no line search is required. Recently, the global convergence for the convex quadratic case has been established. However, for the nonquadratic case, the method needs to be incorporated in a globalization scheme. In this work, a nonmonotone line search strategy that guarantees global convergence is combined with the Barzilai and Borwein method. This strategy is based on the nonmonotone line search technique proposed by Grippo, Lampariello, and Lucidi [SIAM J. Numer. Anal., 23 (1986), pp. 707--716]. Numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented. These results indicate that the global Barzilai and Borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations.

Topics: Gradient method (58%), , Line search (54%)

644 Citations

Book Chapter
Roland Glowinski1, Roland Glowinski2Institutions (2)
Topics: , ,

421 Citations

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