38 research outputs found
Finite Element Methods for Elliptic Distributed Optimal Control Problems with Pointwise State Constraints
Finite element methods for a model elliptic distributed optimal control
problem with pointwise state constraints are considered from the perspective of
fourth order boundary value problems
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Mini-Workshop: Numerical Analysis for Non-Smooth PDE-Constrained Optimal Control Problems
This mini-workshop brought together leading experts working on various aspects of numerical analysis for optimal control problems with nonsmoothness. Fifteen extended abstracts summarize the presentations at this mini-workshop
Discontinuous Galerkin Methods for an Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints
We investigate discontinuous Galerkin methods for an elliptic optimal control
problem with a general state equation and pointwise state constraints on
general polygonal domains. We show that discontinuous Galerkin methods for
general second-order elliptic boundary value problems can be used to solve the
elliptic optimal control problems with pointwise state constraints. We
establish concrete error estimates and numerical experiments are shown to
support the theoretical results
P 1 finite element methods for an elliptic optimal control problem with pointwise state constraints
We present theoretical and numerical results for two P finite element methods for an elliptic distributed optimal control problem on general polygonal/polyhedral domains with pointwise state constraints.
A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints
We consider a one dimensional elliptic distributed optimal control problem
with pointwise constraints on the derivative of the state. By exploiting the
variational inequality satisfied by the derivative of the optimal state, we
obtain higher regularity for the optimal state under appropriate assumptions on
the data. We also solve the optimal control problem as a fourth order
variational inequality by a finite element method, and present the error
analysis together with numerical results
[Formula presented] interior penalty methods for an elliptic state-constrained optimal control problem with Neumann boundary condition
We study [Formula presented] interior penalty methods for an elliptic optimal control problem with pointwise state constraints on two dimensional convex polygonal domains. The approximation of the optimal state is obtained by solving a fourth order variational inequality and the approximation of the optimal control is computed by a post-processing procedure. We prove the convergence of numerical solutions with rates in the [Formula presented]-like energy error by using the complementarity form of the variational inequality. Furthermore, we develop an a posteriori analysis for a residual based error estimator and introduce an adaptive algorithm. Numerical experiments are provided to gauge the performance of the proposed methods
P1 finite element methods for an elliptic state-constrained distributed optimal control problem with Neumann boundary conditions
We investigate two P finite element methods for an elliptic state-constrained distributed optimal control problem with Neumann boundary conditions on general polygonal domains.
Finite Element Methods for One Dimensional Elliptic Distributed Optimal Control Problems with Pointwise Constraints on the Derivative of the State
We investigate finite element methods for one dimensional elliptic
distributed optimal control problems with pointwise constraints on the
derivative of the state formulated as fourth order variational inequalities for
the state variable. For the problem with Dirichlet boundary conditions, we use
an existing regularity result for the optimal state to
derive convergence for the approximation of the
optimal state in the norm. For the problem with mixed Dirichlet and
Neumann boundary conditions, we show that the optimal state belongs to
under appropriate assumptions on the data and obtain convergence for the
approximation of the optimal state in the norm
A cubic C\u3csup\u3e0\u3c/sup\u3e interior penalty method for elliptic distributed optimal control problems with pointwise state and control constraints
We design and analyze a cubic C interior penalty method for linear–quadratic elliptic distributed optimal control problems with pointwise state and control constraints. Numerical results that corroborate the theoretical error estimates are also presented.
Finite Element Methods for Fourth Order Variational Inequalities
In this work we study finite element methods for fourth order variational inequalities. We begin with two model problems that lead to fourth order obstacle problems and a brief survey of finite element methods for these problems. Then we review the fundamental results including Sobolev spaces, existence and uniqueness results of variational inequalities, regularity results for biharmonic problems and fourth order obstacle problems, and finite element methods for the biharmonic problem. In Chapter 2 we also include three types of enriching operators which are useful in the convergence analysis. In Chapter 3 we study finite element methods for the displacement obstacle problem of clamped Kirchhoff plates. A unified convergence analysis is provided for finite element methods, classical nonconforming finite element methods and interior penalty methods. The key ingredient in the error analysis is the introduction of the auxiliary obstacle problem. An optimal error estimate in the energy norm is obtained for convex domains. We also address the approximations of the coincidence set and the free boundary. In Chapter 4 we study a Morley finite element method and a quadratic interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates with general Dirichlet boundary conditions on general polygonal domains. We prove the magnitudes of the errors in the energy norm and the norm are , where is determined by the interior angles of the polygonal domain. Numerical results are also presented to illustrate the performance of the methods and verify the theoretical results obtained in Chapter 3 and Chapter 4. In Chapter 5 we consider an elliptic optimal control problem with state constraints. By formulating the problem as a fourth order obstacle problem with the boundary condition of simply supported plates, we study a quadratic interior penalty method and derive the error estimates in the energy norm based on the framework we introduced in Chapter 3. The rate of convergence is derived for both quasi-uniform meshes and graded meshes. Numerical results presented in this chapter confirm our theoretical results