Journal ArticleDOI
Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses
Max D. Gunzburger,Steven L. Hou +1 more
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In this paper, Lagrange multipliers are used for finite element approximations of the Stokes and Navier-Stokes equations with in-homogeneous essential boundary conditions.Abstract:
Finite element approximations of the Stokes and Navier–Stokes equations with in-homogeneous essential boundary conditions are considered. Boundary conditions are enforced weakly by introducing Lagrange multipliers. Optimal error estimates, including some for the stress vector on the boundary, are derived under minimal regularity assumptions on the data. Particular attention is paid to the analysis of a practical choice of finite element spaces for which the Lagrange multiplier calculation uncouples from that for the velocity and pressure. The results are also applicable to general second-order elliptic systems.read more
Citations
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Journal ArticleDOI
Adaptive Wavelet Methods II—Beyond the Elliptic Case
TL;DR: It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation.
Journal ArticleDOI
Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part II: The Lagrange--Newton Solver and Its Application to Optimal Control of Steady Viscous Flows
George Biros,Omar Ghattas +1 more
TL;DR: LNKS is an order of magnitude faster than quasi-Newton reduced SQP, and it is able to solve previously intractable problems of up to 800,000 state and 5,000 decision variables at about 5 times the cost of a single forward flow solution.
Journal ArticleDOI
Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls
TL;DR: In this article, the Lagrange multiplier technique is used to derive a system of partial differential equations from which optimal solutions may be deduced, and finite element approximations of solutions are defined and optimal error estimâtes are derived.
Journal ArticleDOI
Coupled Generalized Nonlinear Stokes Flow with Flow through a Porous Medium
TL;DR: This article proposes and analyzes an approximation algorithm and establishes a priori error estimates for the approximation and shows existence and uniqueness of a variational solution to the problem.
Journal ArticleDOI
Adjoint Equation-Based Methods for Control Problems in Incompressible, Viscous Flows
TL;DR: A review of adjoint equation-based methodologies for viscous, incompressible flow control and optimization problems is given and illustrated by a drag minimization example.
References
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Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book
Non-homogeneous boundary value problems and applications
TL;DR: In this paper, the authors consider the problem of finding solutions to elliptic boundary value problems in Spaces of Analytic Functions and of Class Mk Generalizations in the case of distributions and Ultra-Distributions.
Book
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
TL;DR: This paper presents the results of an analysis of the "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions and its applications to Mixed Approximation and Homogeneous Stokes Equations.
Book
Navier-Stokes Equations
TL;DR: Schiff's base dichloroacetamides having the formula OR2 PARALLEL HCCl2-C-N ANGLE R1 in which R1 is selected from the group consisting of alkenyl, alkyl, alkynyl and alkoxyalkyl; and R2 is selected by selecting R2 from the groups consisting of lower alkylimino, cyclohexenyl-1 and lower alkynyl substituted cycloenenyl -1 as discussed by the authors.
Journal ArticleDOI
Finite element interpolation of nonsmooth functions satisfying boundary conditions
L. Ridgway Scott,Shangyou Zhang +1 more
TL;DR: In this article, a modified Lagrange type interpolation operator is proposed to approximate functions in Sobolev spaces by continuous piecewise polynomials, and the combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.