Jérôme Lemoine

Bio: Jérôme Lemoine is an academic researcher from University of Auvergne. The author has contributed to research in topics: Order (ring theory) & Newton's method. The author has an hindex of 4, co-authored 9 publications receiving 32 citations.

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

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.

Resolution of the implicit Euler scheme for the Navier–Stokes equation through a least-squares method

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.

A Fully Space-Time Least-Squares Method for the Unsteady Navier–Stokes System

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.

Approximation of null controls for semilinear heat equations using a least-squares approach

Abstract: The null distributed controllability of the semilinear heat equation $\partial_t y-\Delta y + g(y)=f \,1_{\omega}$ assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{\vert r\vert\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fern\'andez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized heat equation. Assuming that $g^\prime$ is bounded and uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in (0,1]$, we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations. The method is based on a least-squares approach and coincides with a globally convergent damped Newton methods: it guarantees the convergence whatever be the initial element of the sequence. Numerical experiments in the one dimensional setting illustrate our analysis.

A fully space-time least-squares method for the unsteady Navier-Stokes system.

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.

Numerical Approximation Of Partial Differential Equations

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Resolution of the implicit Euler scheme for the Navier–Stokes equation through a least-squares method

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.

A Fully Space-Time Least-Squares Method for the Unsteady Navier–Stokes System

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.

Constructive exact control of semilinear 1D wave equations by a least-squares approach

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.

Approximation of null controls for semilinear heat equations using a least-squares approach

Abstract: The null distributed controllability of the semilinear heat equation $\partial_t y-\Delta y + g(y)=f \,1_{\omega}$ assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{\vert r\vert\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fern\'andez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized heat equation. Assuming that $g^\prime$ is bounded and uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in (0,1]$, we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations. The method is based on a least-squares approach and coincides with a globally convergent damped Newton methods: it guarantees the convergence whatever be the initial element of the sequence. Numerical experiments in the one dimensional setting illustrate our analysis.