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

We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The problem arises when attempting to optimize the temperature gradient within crystal growth by the physical vapor transport (PVT) method. Based on a minimum principle for the semilinear equation as well as $L^\infty$-estimates for the weak solution, we establish the existence of an optimal solution as well as necessary optimality conditions. The theoretical results are illustrated by numerical computations.

/pdf/optimal-control-of-a-semilinear-pde-with-nonlocal-radiation-4x9yozr5mt.pdf

Optimal Control of a Semilinear PDE with Nonlocal Radiation Interface Conditions

Solutions to optimization problems with pde constraints inherit special properties; the associated state solves the pde which in the optimization problem takes the role of a equality constraint, and this state together with the associated control solves an optimization problem, i.e. together with multipliers satisfies first and second order necessary optimality conditions. In this note we review the state of the art in designing discrete concepts for optimization problems with pde constraints with emphasis on structure conservation of solutions on the discrete level, and on error analysis for the discrete variables involved. As model problem for the state we consider an elliptic pde which is well understood from the analytical point of view. This allows to focus on structural aspects in discretization. We discuss the approaches First discretize, then optimize and First optimize, then discretize, and consider in detail two variants of the First discretize, then optimize approach, namely variational discretization, a discrete concept which avoids explicit discretization of the controls, and piecewise constant control approximations. We consider general constraints on the control, and also consider pointwise bounds on the state. We outline the basic ideas for providing optimal error analysis and complement our analytical findings with numerical examples which confirm our analytical results (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

http://page.math.tu-berlin.de/~troeltz/arbeiten/Hinze_Troeltzsch.pdf

Discrete concepts versus error analysis in pde constrained optimization

In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal solutions of the problem.

Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations

In this paper we study Lavrentiev-type regularization concepts for linear-quadratic parabolic control problems with pointwise state constraints. In the first part, we apply classical Lavrentiev regularization to a problem with distributed control, whereas in the second part, a Lavrentiev-type regularization method based on the adjoint operator is applied to boundary control problems with state constraints in the whole domain. The analysis for both classes of control problems is investigated and numerical tests are conducted. Moreover the method is compared with other numerical techniques.

/pdf/on-regularization-methods-for-the-numerical-solution-of-2hrf7p2adl.pdf

On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints

An optimal control problem for a semilinear parabolic equation is investigated, where pointwise constraints are given on the control and the state. The state constraints are of mixed (bottleneck) type, where associated Lagrange multipliers can be assumed to be bounded and measurable functions. Based on this property, a second-order sufficient optimality condition is established that considers strongly active constraints.

Fredi Tröltzsch

Papers

Optimal Control of a Semilinear PDE with Nonlocal Radiation Interface Conditions

Discrete concepts versus error analysis in pde constrained optimization

Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations

On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints

Sufficient Second-Order Optimality Conditions for a Parabolic Optimal Control Problem with Pointwise Control-State Constraints