A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Finite element methods for Maxwell's equations.

Thank you for downloading linear and nonlinear programming. As you may know, people have search numerous times for their favorite novels like this linear and nonlinear programming, but end up in malicious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they juggled with some infectious bugs inside their desktop computer. linear and nonlinear programming is available in our book collection an online access to it is set as public so you can download it instantly. Our digital library spans in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the linear and nonlinear programming is universally compatible with any devices to read.

Linear And Nonlinear Programming

It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction-diffusion systems which are — on the one h and — typical for the whole class of systems to which the general theory applies and — on the other h and — still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.

Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems

Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)

Convection Heat Transfer

Die mathematische Theorie der optimalen Steuerung hat sich im Zusammenhang mit Berechnungen fur die Raumfahrt schnell zu einem wichtigen und eigenstandigen Gebiet der angewandten Mathematik entwickelt. Die Bewegungsgleichungen von Luft- und Raumfahrzeugen werden durch Systeme von gewohnlichen Differentialgleichungen beschrieben und Aspekte der Optimierung kommen dann ins Spiel, wenn die Bewegungen der Flugkorper optimal ablaufen sollen.

Einführung und Beispiele

We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuska's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.

Space-time finite element discretization of parabolic optimal control problems with energy regularization

In this paper a numerical method is presented to solve a state-constrained parabolic boundary control problem. The optimization technique is based on generalized bang-bang-properties of the optimal control, which were proved in earlier papers of the author. The effectivity of the corresponding algorithm is demonstrated by means of two examples, which were solved with high accuracy.



In dieser Arbeit wird eine numerische Methode fur die Losung eines parabolischen Randsteuerproblems mit Zustandsbeschrankungen vorgestellt. Die Optimierungstechnik nutzt verallgemeinerte Bang-Bang-Eigenschaften der optimalen Steuerung aus, welche in fruheren Arbeiten des Autors bewiesen wurden. Anhand von zwei Beispielen, die mit hoher Genauigkeit gelost wurden, zeigt sich die Effektivitat des Verfahrens.

The generalized bang-bang-principle and the numerical solution of a parabolic boundary-control problem with constraints on the control and the state

An optimal control problem for a semilinear heat equation with distributed control is discussed, where two-sided pointwise box constraints on the control and two-sided pointwise mixed control-state constraints are given. The objective functional is the sum of a standard quadratic tracking type part and a multiple of the $L^1$-norm of the control that accounts for sparsity. Under a certain structural condition on almost active sets of the optimal solution, the existence of integrable Lagrange multipliers is proved for all inequality constraints. For this purpose, a theorem by Yosida and Hewitt is used. It is shown that the structural condition is fulfilled for all sufficiently large sparsity parameters. The sparsity of the optimal control is investigated. Eventually, higher smoothness of Lagrange multipliers is shown up to H\"older regularity.

/pdf/sparse-optimal-control-for-a-semilinear-heat-equation-with-2c5u23fq5y.pdf

Sparse optimal control for a semilinear heat equation with mixed control-state constraints – regularity of Lagrange multipliers

/pdf/problems-and-methods-in-mathematical-physics-5a9mhr6tmp.pdf

Fredi Tröltzsch

Papers

Einführung und Beispiele

Space-time finite element discretization of parabolic optimal control problems with energy regularization

The generalized bang-bang-principle and the numerical solution of a parabolic boundary-control problem with constraints on the control and the state

Sparse optimal control for a semilinear heat equation with mixed control-state constraints – regularity of Lagrange multipliers

Problems and Methods in Mathematical Physics