Mixed Finite Element Method for Dirichlet Boundary Control Problem Governed by Elliptic PDEs

Abstract The parabolic Dirichlet boundary control problem and its finite element discretization are considered in convex polygonal and polyhedral domains. We improve the existing results on the regularity of the solutions by establishing and utilizing the maximal $L^p$-regularity of parabolic equations under inhomogeneous Dirichlet boundary conditions. Based on the proved regularity of the solutions, we prove ${\mathcal O}(h^{1-1/q_0-\epsilon })$ convergence for the semidiscrete finite element solutions for some $q_0>2$, with $q_0$ depending on the maximal interior angle at the corners and edges of the domain and $\epsilon$ being a positive number that can be arbitrarily small.

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A characteristic-mixed finite element method for time-dependent convection–diffusion optimal control problem

Applied Mathematics and Computation ◽

10.1016/j.amc.2011.08.087 ◽

2011 ◽

Vol 218 (7) ◽

pp. 3430-3440 ◽

Cited By ~ 5

Author(s):

Hongfei Fu ◽

Hongxing Rui

Keyword(s):

Finite Element Method ◽

Optimal Control ◽

Finite Element ◽

Optimal Control Problem ◽

Control Problem ◽

Mixed Finite Element ◽

Mixed Finite Element Method ◽

Time Dependent ◽

Convection Diffusion ◽

Element Method

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A Finite Element Method for Elliptic Dirichlet Boundary Control Problems

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2019-0104 ◽

2020 ◽

Vol 20 (4) ◽

pp. 827-843

Author(s):

Michael Karkulik

Keyword(s):

Finite Element ◽

Boundary Control ◽

Dirichlet Boundary ◽

Mixed Finite Element ◽

Lagrangian Multiplier ◽

Finite Element Discretization ◽

Dirichlet Boundary Control ◽

Element Discretization ◽

Polygonal Domains ◽

Constrained Problems

AbstractWe consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in {H^{1/2}(\Gamma)}. To avoid computing the latter norm numerically, we realize it using the {H^{1}(\Omega)} norm of the harmonic extension of the control. We propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier. In the case of convex polygonal domains, optimal error estimates in the {H^{1}} and {L_{2}} norm are proven. We also consider and analyze the case of control constrained problems.

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Crank-Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problem

10.22541/au.163250059.90676201/v1 ◽

2021 ◽

Author(s):

Caijie Yang ◽

Tongjun Sun

Keyword(s):

Finite Difference ◽

Control Problem ◽

Boundary Control ◽

Dirichlet Boundary ◽

Second Order ◽

Difference Schemes ◽

Optimality System ◽

Finite Difference Schemes ◽

Dirichlet Boundary Control ◽

Boundary Control Problem

In this paper, we adopt the optimize-then-discretize approach to solve parabolic optimal Dirichlet boundary control problem. First, we derive the first-order necessary optimality system, which includes the state, co-state equations and the optimality condition. Then, we propose Crank-Nicolson finite difference schemes to discretize the optimality system in 1D and 2D cases, respectively. In order to build the second order spatial approximation, we use the ghost points on the boundary in the schemes. We prove that the proposed schemes are unconditionally stable, compatible and second-order convergent in both time and space. To avoid solving the large coupled schemes directly, we use the iterative method. Finally, we present a numerical example to validate our theoretical analysis.

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