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

Optimal control of semilinear parabolic equations

In this paper we deal with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition. The case of a boundary control is studied. We consider pointwise constraints on the control and certain equality and inequality constraints on the state. Second order sufficient conditions for local optimality of controls are derived.

/pdf/second-order-sufficient-optimality-conditions-for-a-class-of-2q0vzye6cx.pdf

Second order sufficient optimality conditions for a class of elliptic control problems

We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a non-differentiable term with the measure-norm of the control. Existence, uniqueness and regularity of the solution of the state equation, as well as differentiability properties of the control-to-state operator are obtained. Next, we provide first order optimality conditions for local solutions. Finally, the control space is suitably discretized and we prove convergence of the solutions of the discrete problems to the solutions of the original problem. Several numerical examples are included to illustrate the theoretical results.

/pdf/measure-control-of-a-semilinear-parabolic-equation-with-a-2g1mg0h4iq.pdf

Measure Control of a Semilinear Parabolic Equation with a Nonlocal Time Delay

Current research in the control of PDEs is focused on highly nonlinear coupled systems of partial differential equations that arise from diverse applications in engineering and science. Dealing with associated control problems calls for a careful analysis of such systems, efficient numerical methods for differential equations and powerful techniques of numerical optimization. The program of the conference contained a blend of associated talks. Systems modelling quantum effects, dynamic fluid-structure interaction, the coupling of heat transport or fluid flow with electromagnetic fields and compressible flows were subject of the talks. Main aspects of control theory were stateconstrained optimal control, mesh-adaptivity and a posteriori error estimation, feedback control, free material and shape optimization, controllability and observability. The conference tightened the links between applications, numerics, and analysis with some emphasis on the analytic aspects. Mathematics Subject Classification (2000): 49J20,35Qxx,65K05. Introduction by the Organisers The international conference Optimal Control of Coupled Systems of PDE, was held March 2nd–March 8th, 2008, organized by K. Kunisch (Karl-Franzens-University Graz), G. Leugering (University of Erlangen-Nurnberg), J. Sprekels (Weierstrass Institute of Applied Analysis and Stochastics Berlin) and Fredi Troltzsch (Technische Universitat Berlin). 44 participants attended this meeting and followed 33 talks on optimal control and related topics. Mathematically, the control of partial differential equations (PDEs) is concerned with the following type of problems: The solution of a PDE (the state of the system) should be influenced in a desired way by the choice of certain control functions or control parameters (the controls), which may occur in different terms 588 Oberwolfach Report 13/2008 of the differential equation. If the controls are to minimize a certain functional related to the state, then an optimal control problem is posed. If the domain underlying the PDE is subject of the control, then a shape optimization problem is given. For evolution equations, it can be required to move the solution from a given initial state exactly to a desired final state. This is the question of exact controllability. Optimization and control of partial differential equations continues to be a very active field of research. Scientists working in different fields came together to report on their contributions to the numerical analysis of control problems. It is remarkable that optimal control is a challenge for researchers with backgrounds in related fields such as the theory of systems of nonlinear PDEs, numerical methods for solving them, large scale nonlinear optimization, or the numerical simulation and optimization of complex processes in engineering or medical science. This diversity was reflected by the conference program. Talks were focused on • applications of optimal control to the thermistor problem, crystal growth, quantum mechanics or aviation • state-constrained optimal control problems • controllability and observability of the Navier-Stokes equations and of systems for fluid-structure interaction; feedback control • Hamilton-Bellman-Jacobi equations • models for the interaction of electromagnetic fields, heat transport and fluid flow • mesh adaptivity, a-posteriori and a-priori error estimates for the solutions of optimal control problems • the application of numerical techniques such as semismooth Newton methods, multilevel techniques or domain decomposition • firstand second-order optimality conditions for the optimal controls of nonlinear systems of PDEs arising from different applications • modal control • the optimal shape design of electromagnetic systems or thin shells and on free material optimization. All these issues are currently subject of active research. In extensive and lively discussions, the participants of the workshop produced new mathematical ideas and tightened connections of joint cooperation.

Optimal Control of Coupled Systems of PDE

Control of Distributed Parameter Systems -- Stochastic Optimization and Control -- Stabilization, Feedback, and Model Predictive Control -- Flow Control -- Shape and Structural Optimization -- Applications and Control of Lumped Parameter Systems.

Fredi Tröltzsch

Papers

Optimal control of semilinear parabolic equations

Second order sufficient optimality conditions for a class of elliptic control problems

Measure Control of a Semilinear Parabolic Equation with a Nonlocal Time Delay

Optimal Control of Coupled Systems of PDE

System Modeling and Optimization 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers