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 the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: Let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated. Resume. ... 1991 Mathematics Subject Classification. 49K20, 35J61, 35K58. The dates will be set by the publisher. Introduction In this paper, we focus on the following question for a class of optimal control problems for semilinear parabolic equations: Suppose that a numerical approximation ũ for a locally optimal control is given. For instance, it might have been obtained by a numerical optimization method or as the solution to some reduced order optimization model. How far is this control from the nearest locally optimal control ū? Of course, we have to assume that such a solution ū exists in a neighborhood of ũ. Moreover, ũ should already be sufficiently close to ū. The question is to quantify the error ‖ũ− ū‖ in an appropriate norm. In principle, this is not a paper on proper orthogonal decomposition (POD). Our estimation method is not restricted to any specific method of numerical approximation for ũ. Our primary goal is a numerical application of a perturbation method used by Arada et al. [2] in the context of FEM approximations of elliptic optimal control problems. The main idea of this method goes back to Dontchev et al. [7] and Malanowski and Maurer [18], who introduced it for the a priori error analysis of optimal control problems governed by ordinary differential equations.

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A method of a-posteriori error estimation with application to proper orthogonal decomposition

On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints

In this work, boundary control problems governed by a system of semilinear parabolic PDEs with pointwise control constraints are considered. This class of problems is related to applications in the chemical catalysis. After discussing existence and uniqueness of the state equation with both linear and nonlinear boundary conditions, the existence of an optimal solution is shown. Necessary and sufficient optimality conditions are derived to deal with numerical examples, which conclude the paper.

/pdf/optimal-boundary-control-of-a-system-of-reaction-diffusion-1p6vqnpcmf.pdf

Optimal boundary control of a system of reaction diffusion equations

This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuska'...

Unstructured Space-Time Finite Element Methods for Optimal Control of Parabolic Equations

Discretizations of optimal control problems for elliptic equations by finite element methods are considered The problems are subject to constraints on the control and may also contain pointwise state constraints Some techniques are surveyed to estimate the distance between the exact optimal control and the associated optimal control of the discretized problem As a particular example, an error estimate for a nonlinear optimal control problem with finitely many control values and state constraints in finitely many points of the spatial domain is derived.

/pdf/on-finite-element-error-estimates-for-optimal-control-3n2zk6qxra.pdf

Fredi Tröltzsch

Papers

A method of a-posteriori error estimation with application to proper orthogonal decomposition

On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints

Optimal boundary control of a system of reaction diffusion equations

Unstructured Space-Time Finite Element Methods for Optimal Control of Parabolic Equations

On finite element error estimates for optimal control problems with elliptic PDEs