Error bounds for a Dirichlet boundary control problem based on energy spaces
TLDR
A priori error estimates of optimal order in the energy norm and the L-2-norm are derived and a reliable and efficient a posteriori error estimator is derived with the help of an auxiliary problem.Abstract:
In this article, an alternative energy-space based approach is proposed for the Dirichlet boundary control problem and then a finite-element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the L-2-norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help of an auxiliary problem. The theoretical results are illustrated by the numerical experiments.read more
Citations
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Journal ArticleDOI
A convergent adaptive finite element method for elliptic Dirichlet boundary control problems
TL;DR: It is proved that the sequence of adaptively generated discrete solutions including the control, the state and the adjoint state, guided by the newly derived a posteriori error indicators, converges to the true solution along with the convergence of the error estimators.
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A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs
TL;DR: An HDG method is proposed for a Dirichlet boundary control problem for the Poisson equation, and optimal a priori error estimates for the control are obtained.
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A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity
TL;DR: In this paper, the authors proposed a hybridizable discontinuous Galerkin (HDG) method to approximate the Dirichlet boundary control problem for elliptic convection diffusion PDE and proved an optimal superlinear convergence rate for the control under certain assumptions on the domain and on the target state.
Journal ArticleDOI
Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system
TL;DR: A hybridizable discontinuous Galerkin (HDG) method is proposed and analyzed to approximate the solution of an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an L 2 penalty, and the theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary.
Journal ArticleDOI
Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization
TL;DR: The regularity of the solution is carefully carved out exploiting weighted Sobolev and Hölder spaces, allowing to derive a sharp relation between the convergence rates for the approximation and the structure of the geometry, more precisely, the largest opening angle at the vertices of polygonal domains.
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
Finite Element Method for Elliptic Problems
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Book
The Mathematical Theory of Finite Element Methods
TL;DR: In this article, the construction of a finite element of space in Sobolev spaces has been studied in the context of operator-interpolation theory in n-dimensional variational problems.
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
Elliptic Problems in Nonsmooth Domains
TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.