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

nonlinear functional analysis and its applications is available in our book collection an online access to it is set as public so you can get it instantly. Our books collection hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the nonlinear functional analysis and its applications is universally compatible with any devices to read.

Nonlinear Functional Analysis And Its Applications

The Projection Theorem. Theorems on Extension and Separation. Dual Spaces and Transposed Operators. The Banach Theorem and the Banach-Steinhaus Theorem. Construction of Hilbert Spaces. L ~ 2 Spaces and Convolution Operators. Sobolev Spaces of Functions of One Variable. Some Approximation Procedures in Spaces of Functions. Sobolev Spaces of Functions of Several Variables and the Fourier Transform. Introduction to Set-Valued Analysis and Convex Analysis. Elementary Spectral Theory. Hilbert-Schmidt Operators and Tensor Products. Boundary Value Problems. Differential-Operational Equations and Semigroups of Operators. Viability Kernels and Capture Basins. First-Order Partial Differential Equations. Selection of Results. Exercises. Bibliography. Index.

Applied functional analysis, by Jean-Pierre Aubin; translated by Carole Labrousse. Pp 423. £16·50. 1980. ISBN 0-471-02149-0 (Wiley)

Thank you very much for reading applied functional analysis. As you may know, people have look numerous times for their favorite books like this applied functional analysis, but end up in infectious downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they cope with some infectious virus inside their computer. applied functional analysis is available in our book collection an online access to it is set as public so you can get it instantly. Our digital library saves in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the applied functional analysis is universally compatible with any devices to read.

Applied Functional Analysis

Free And Moving Boundary Problems

This paper is concerned with the distributed optimal control of a time-discrete Cahn--Hilliard--Navier--Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier--Stokes equation. By proposing a suitable time-discretization, energy estimates are proved, and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The latter correspond to Moreau--Yosida-type approximations of the double-obstacle potential. The consistency of these approximations is shown, and first-order optimality conditions for the regularized problems are derived. Through a limit process with respect to the regularization parameter, a stationarity system for the original problem is established. The resulting system corresponds to a function space version of C-s...

Optimal Control of a Semidiscrete Cahn--Hilliard--Navier--Stokes System with Nonmatched Fluid Densities

This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn–Hilliard–Navier–Stokes system with variable densities. The free energy density associated with the Cahn–Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier–Stokes equation. A dual-weighted residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residuals weighted by approximate dual quantities and vice versa as well as various complementarity mismatch errors. Details of the numerical realization of the adaptive concept and a report on the numerical tests are given.

A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system

This paper is concerned with the distributed optimal control of a time-discrete Cahn--Hilliard/Navier--Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instance of a variational inequality of fourth order and the Navier--Stokes equation. By proposing a suitable time-discretization, energy estimates are proved and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The latter correspond to Moreau--Yosida type approximations of the double-obstacle potential. The consistency of these approximations is shown and first order optimality conditions for the regularized problems are derived. Through a limit process, a stationarity system for the original problem is established which is related to a function space version of C-stationarity.

Optimal Control of a Semidiscrete Cahn-Hilliard-Navier-Stokes System with Non-Matched Fluid Densities

A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn--Hilliard--Navier--Stokes system

Optimal control problems for geometric (evolutionary) partial differential inclusions are considered. The focus is on problems which, in addition to the nonlinearity due to geometric evolution, contain optimization theoretic challenges because of non-smoothness. The latter might stem from energies containing non-smooth constituents such as obstacle-type potentials or terms modeling, e.g., pinning phenomena in microfluidics. Several techniques to remedy the resulting constraint degeneracy when deriving stationarity conditions are presented. A particular focus is on Yosida-type mollifications approximating the original degenerate problem by a sequence of non-degenerate nonconvex optimal control problems. This technique is also the starting point for the development of numerical solution schemes. In this context, also dual-weighted residual based error estimates are addressed to facilitate an adaptive mesh refinement. Concerning the underlying state model, sharp and diffuse interface formulations are discussed. While the former always allows for accurately tracing interfacial motion, the latter model may be dictated by the underlying physical phenomenon, where near the interface mixed phases may exist, but it may also be used as an approximate model for (sharp) interface motion. In view of the latter, (sharp interface) limits of diffuse interface models are addressed. For the sake of presentation, this exposition confines itself to phase field type diffuse interface models and, moreover, develops the optimal control of either of the two interface models along model applications. More precisely, electro-wetting on dielectric is used in the sharp interface context, and the control of multiphase fluids involving spinodal decomposition highlights the phase field technique. Mathematically, the former leads to a Hele-Shaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a Cahn-Hilliard Navier-Stokes model including a non-smooth obstacle type potential leading to a variational inequality constraint.

Tobias Keil

Papers

Optimal Control of a Semidiscrete Cahn--Hilliard--Navier--Stokes System with Nonmatched Fluid Densities

A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system

Optimal Control of a Semidiscrete Cahn-Hilliard-Navier-Stokes System with Non-Matched Fluid Densities

A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn--Hilliard--Navier--Stokes system

Optimal control of geometric partial differential equations