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.

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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

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Erratum to: On the optimal control of the Schlögl-model

A class of Pyragas type nonlocal feedback controllers with time-delay is investigated for the Schlogl model. The main goal is to find an optimal kernel in the controller such that the associated solution of the controlled equation is as close as possible to a desired spatio-temporal pattern. An optimal kernel is the solution to a nonlinear optimal control problem with the kernel taken as control function. The well-posedness of the optimal control problem and necessary optimality conditions are discussed for different types of kernels. Special emphasis is laid on time-periodic functions as desired patterns. Here, the cross correlation between the state and the desired pattern is invoked to set up an associated objective functional that is to be minimized. Numerical examples are presented for the 1D Schlogl model and a class of simple step functions for the kernel.

Optimization of nonlocal time-delayed feedback controllers

In this paper a regularity condition for the non-linear mathematical programming problem in Banach spaces is introduced, which is a stronger version of a condition published by J. Zowe and S. Kurcyusz in 1979. The corresponding Lagrange multiplier rule is applied to prove necessary optimality conditions for nonlinear optimal control problems of Noether operator equations with constraints on the control and the state.

A modification of the zowe and kurcyusz regularity condition with application to the optimal control of noether operator equations with constraints on the control and the state

We define a class of parabolic problems with control and state constraints and identify a problem within this class which possesses a locally unique critical point satisfying the second order sufficient optimality conditions In this solution inequality constraints on the control are strongly active The second derivative of the Lagrangian is not globally coercive This is both shown analytically as well as verified numerically for a finite difference discretization

Sufficient Optimality in a Parabolic Control Problem

A state-constrained optimal boundary control problem governed by a linear elliptic equation is considered. In order to obtain the optimality conditions for the solutions to the model problem, a Slater assumption has to be made that restricts the theory to the two-dimensional case. This difficulty is overcome by a source representation of the control and combined with a Lavrentiev type regularization. Optimality conditions for the regularized problem are derived, where the corresponding Lagrange multipliers have L2-regularity. By the spectral theorem for compact and normal operators, the convergence result of [22] is extended to a higher dimensional case. Moreover, the convergence for vanishing regularization parameter of the adjoint state associated with the regularized problem is shown. Finally, the uniform boundedness of the regularized Lagrange multipliers in L1(Ω) is verified by a maximum principle argument.

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Fredi Tröltzsch

Papers

Erratum to: On the optimal control of the Schlögl-model

Optimization of nonlocal time-delayed feedback controllers

A modification of the zowe and kurcyusz regularity condition with application to the optimal control of noether operator equations with constraints on the control and the state

Sufficient Optimality in a Parabolic Control Problem

Source Representation Strategy for Optimal Boundary Control Problems with State Constraints