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A fully space-time least-squares method for the unsteady Navier-Stokes system.
TL;DR: A minimizing sequence for the least-squares functional is constructed which converges strongly to a solution of the Navier-Stokes system and the convergence is quadratic.
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.
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TL;DR: 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.
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.
11 citations
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TL;DR: In this paper, a constructive proof and algorithm for the controllability of semilinear 1D wave equations with Dirichlet boundary conditions is presented. But the proof is based on the Leray-Schauder fixed point theorem, which is not constructive.
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.
8 citations
Cites methods from "A fully space-time least-squares me..."
...th linear order under general assumptions (see [6, Theorem 8.7]). As far as we know, damped type Newton methods have been little applied to partial differential equations in the literature. We mention [9, 14] in the context of fluid mechanics. 14 Another variant. To simplify, let us take λk = 1, as in the standard Newton method. Then, for each k∈ N, the optimal pair (Y1 k ,F 1 k ) ∈ A0 is such that the ele...
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TL;DR: The null distributed controllability of the semilinear heat equation was shown to be super linear with a rate equal to 1 + s in this paper, where s is the initial element of the sequence.
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.
4 citations
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TL;DR: This general method provides a constructive proof of the exact controllability for the semilinear wave equation, based on a least-squares approach, which guarantees the convergence whatever the initial element of the sequence may be.
Abstract: The exact distributed controllability of the semilinear wave equation ytt − yxx + g(y) = f 1ω, assuming that g satisfies the growth condition |g(s)|/(|s| log 2 (|s|)) → 0 as |s| → ∞ and that g ∈ L ∞ loc (R) has been obtained by Zuazua in the nineties. The proof based on a Leray-Schauder fixed point argument makes use of precise estimates of the observability constant for a linearized wave equation. It does not provide however an explicit construction of a null control. Assuming that g ∈ L ∞ loc (R), that sup a,b∈R,a =b |g (a) − g (b)|/|a − b| r < ∞ for some r ∈ (0, 1] and that g satisfies the growth condition |g (s)|/ log 2 (|s|) → 0 as |s| → ∞, 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 guarantees the convergence whatever the initial element of the sequence may be. In particular, after a finite number of iterations, the convergence is super linear with rate 1 + r. This general method provides a constructive proof of the exact controllability for the semilinear wave equation.
Cites background from "A fully space-time least-squares me..."
...We mention [9, 14] in the context of fluids mechanics....
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...Proceeding as in [9], we are now in position to prove the strong convergence result for the sequences (E(yk, fk))(k≥0) and (yk, fk)(k≥0) for the norm ‖ · ‖A....
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TL;DR: In this article , a global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space-time multigrid algorithm.
Abstract: Abstract In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.
References
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Book•
26 Feb 1977
TL;DR: Schiff's base dichloroacetamides having the formula OR2 PARALLEL HCCl2-C-N ANGLE R1 in which R1 is selected from the group consisting of alkenyl, alkyl, alkynyl and alkoxyalkyl; and R2 is selected by selecting R2 from the groups consisting of lower alkylimino, cyclohexenyl-1 and lower alkynyl substituted cycloenenyl -1 as discussed by the authors.
Abstract: Schiff's base dichloroacetamides having the formula OR2 PARALLEL HCCl2-C-N ANGLE R1 in which R1 is selected from the group consisting of alkenyl, alkyl, alkynyl and alkoxyalkyl; and R2 is selected from the group consisting of alkenyl-1, lower alkylimino, cyclohexenyl-1 and lower alkyl substituted cyclohexenyl-1. The compounds of this invention are useful as herbicidal antidotes.
4,252 citations
TL;DR: First the freefem++ software deals with mesh adaptation for problems in two and three dimension, second, it solves numerically a problem with phase change and natural convection, and finally to show the possibilities for HPC the software solves a Laplace equation by a Schwarz domain decomposition problem on parallel computer.
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.
2,867 citations
"A fully space-time least-squares me..." refers methods in this paper
...We report some numerical results performed with the FreeFem++ package developed at Sorbonne university (see [9])....
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Book•
01 Aug 1994
TL;DR: In this article, the authors provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation.
Abstract: This is the softcover reprint of the very popular hardcover edition. This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method.
2,383 citations
Journal Article•
874 citations
"A fully space-time least-squares me..." refers background in this paper
...timate. Lemma 2.1. Let any u2H, v;w2V . There exists a constant c= c( ) such that (2.1) Z urvwckuk Hkvk V kwk V : Proof. If u2H, v;w2V , denoting ~u;v~ and ~wtheir extension to 0 in R2, we have, see [4] and [20] Z urvw = Z ~u r~vw~ ku~ rv~k H1(R2)kw~k BMO(R2) ck~uk 2kr~vkkw~k H1(R2) ckuk 2krvk 2kwk H1( )2 ckuk Hkvk V kwk V : Lemma 2.2. Let any u2L1(0;T;H) and v2L2(0;T;V ). Then the function B(u;...
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