Dirichlet Boundary Control Problems Research Articles

Error analysis for the finite element approximations of Dirichlet parabolic boundary control problem with measure data

This article investigates the convergence analysis of finite element method for Dirichlet boundary control problem governed by parabolic equation with measure data. The presence of measure data in the source term and Dirichlet boundary control lead the solution of the state equation to have low regularity, which creates a challenging task for the error analysis. We prove the existence, uniqueness and regularity results of the solution to the control problem. For the discretization of the state and adjoint variables, we employ continuous piecewise linear polynomials while the control variable is approximated by piecewise constant functions. The backward Euler technique provides the foundation for the temporal discretization. For the completely discrete optimal control problem, a priori error bounds for the state variable in the L2(0,T;L2(Ω))-norm and for the control variable in the L2(0,T;L2(∂Ω))-norm are derived. Numerical experiment is performed to validate our theoretical analysis.

Modified [formula omitted] interior penalty analysis for fourth order Dirichlet boundary control problem and a posteriori error estimates

We revisit the L2 norm error estimate for the C0-interior penalty analysis of fourth order Dirichlet boundary control problem. The L2 norm error estimate for the optimal control is derived under reduced regularity assumption and this analysis can be carried out on any convex polygonal domain. Moreover, residual based a posteriori error bounds are derived for the optimal control, state and adjoint state variables under minimal regularity assumptions. The error estimator is shown to be reliable and locally efficient. The theoretical findings are illustrated by numerical experiments.

Virtual Element Method for Control Constrained Dirichlet Boundary Control Problem Governed by the Diffusion Problem

Virtual Element Method for Control Constrained Dirichlet Boundary Control Problem Governed by the Diffusion Problem

Optimal control of lake eutrophication

We consider an optimal control problem for lake eutrophication. The state problem is given by coupled nonlinear partial differential equations ruling the dynamics of phosphorus stock and cyanobacteria concentration. Our control is only defined on the lake shore, thus leading to a Dirichlet boundary control problem. The well-posedness of the state problem is proved. The existence of a global solution for the optimal control is the main result of this paper.

Nitsche’s method for elliptic Dirichlet boundary control problems on curved domains

Nitsche’s method for elliptic Dirichlet boundary control problems on curved domains

Mixed finite element method for a second order Dirichlet boundary control problem

The main aim of this article is to analyze mixed finite element method for the second order Dirichlet boundary control problem. We develop both a priori and a posteriori error analysis using the energy space based approach. We obtain optimal order a priori error estimates in the energy norm and L2-norm with the help of auxiliary problems. The reliability and the efficiency of proposed a posteriori error estimator is discussed using the Helmholtz decomposition. Numerical experiments are presented to confirm the theoretical findings.

A two level finite element method for Stokes constrained Dirichlet boundary control problem

In this paper we present a finite element analysis for a Dirichlet boundary control problem governed by the Stokes equation. The Dirichlet control is considered in a convex closed subset of the energy space $\mathbf{H}^1(\Omega).$ Most of the previous works on the Stokes Dirichlet boundary control problem deals with either tangential control or the case where the flux of the control is zero. This choice of the control is very particular and their choice of the formulation leads to the control with limited regularity. To overcome this difficulty, we introduce the Stokes problem with outflow condition and the control acts on the Dirichlet boundary only hence our control is more general and it has both the tangential and normal components. We prove well-posedness and discuss on the regularity of the control problem. The first-order optimality condition for the control leads to a Signorini problem. We develop a two-level finite element discretization by using $\mathbf{P}_1$ elements(on the fine mesh) for the velocity and the control variable and $P_0$ elements (on the coarse mesh) for the pressure variable. The standard energy error analysis gives $\frac{1}{2}+\frac{\delta}{2}$ order of convergence when the control is in $\mathbf{H}^{\frac{3}{2}+\delta}(\Omega)$ space. Here we have improved it to $\frac{1}{2}+\delta,$ which is optimal. Also, when the control lies in less regular space we derived optimal order of convergence up to the regularity. The theoretical results are corroborated by a variety of numerical tests.

Finite Element Analysis of a Constrained Dirichlet Boundary Control Problem Governed by a Linear Parabolic Equation

Finite Element Analysis of a Constrained Dirichlet Boundary Control Problem Governed by a Linear Parabolic Equation

Unified discontinuous Galerkin finite element methods for second order Dirichlet boundary control problem

In this article, we study the Dirichlet boundary control problem governed by Poisson equation, therein the control is penalized in H1(Ω) space and various symmetric discontinuous Galerkin finite element methods are designed and analyzed for its numerical approximation. Symmetric property of the bilinear forms is exploited to obtain the discrete optimality system. By a careful use of various intermediate problems, the optimal order convergence rates are obtained for the control in the energy and L2-norms. Moreover, using an auxiliary system of equations, a posteriori error estimator is derived which is shown to be reliable and efficient. Numerical experiment results are included to confirm the theoretical findings.

Crank–Nicolson finite difference schemes for parabolic optimal Dirichlet boundary control problems

In this paper, we adopt the optimize‐then‐discretize approach to solve parabolic optimal Dirichlet boundary control problems. 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 several numerical examples to validate our theoretical analysis.

Analysis and Approximations of Dirichlet Boundary Control of Stokes Flows in the Energy Space

We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the $L^2$ norm and an energy space seminorm. We prove well-posedness and regularity results for both problems, develop finite element discretizations for both problems, and prove finite element error estimates for the latter problem. The motivation to study the energy space problem follows from our analysis: we prove that the choice of the control space ${\bm L}^2(\Gamma)$ can lead to an optimal control with discontinuities at the corners, even when the domain is convex. We observe this phenomenon in numerical experiments. This behavior does not occur in Dirichlet boundary control problems for the Poisson equation on convex polygonal domains, and may not be desirable in real applications. For the energy space problem, we derive the first order optimality conditions, and show that the solution of the control problem is more regular than the solution of the problem with the ${\bm L}^2(\Gamma)$ regularization. We also prove a priori error estimates for the control in the energy norm, and present several numerical experiments for both control problems on convex and nonconvex domains.

The SIPG method of Dirichlet boundary optimal control problems with weakly imposed boundary conditions

<abstract><p>In this paper, we consider the symmetric interior penalty Galerkin (SIPG) method which is one of Discontinuous Galerkin Methods for the Dirichlet optimal control problems governed by linear advection-diffusion-reaction equation on a convex polygonal domain and the difficulties which we faced while solving this problem numerically. Since standard Galerkin methods have failed when the boundary layers have occurred and advection diffusion has dominated, these difficulties can occur in the cases of higher order elements and non smooth Dirichlet data in using standard finite elements. We find the most convenient natural setting of Dirichlet boundary control problem for the Laplacian and the advection diffusion reaction equations.After converting the continuous problem to an optimization problem, we solve it by "discretize-then-optimize" approach. In final, we estimate the optimal priori error estimates in suitable norms of the solutions and then support the result and the features of the method with numerical examples on the different kinds of domain.</p></abstract>

A finite element method for Dirichlet boundary control of elliptic partial differential equations

This paper introduces a new variational formulation for Dirichlet boundary control problem of elliptic partial differential equations, based on observations that the state and adjoint state are related through the control on the boundary of the domain, and that such a relation may be imposed in the variational formulation of the adjoint state. Well-posedness (unique solvability and stability) of the variational problem is established in the $H^{1}(\Omega)\times H_{0}^{1}(\Omega)$ space for the respective state and adjoint state. A finite element method based on this formulation is analyzed. It is shown that the conforming $k-$th order finite element approximations to the state and the adjoint state, in the respective $L^{2}$ and $H^{1}$ norms converge at the rate of order $k-1/2$ on quasi-uniform mesh for conforming element of order $k$. Numerical examples are presented to validate the theory.

Residual based a posteriori error estimation for Dirichlet boundary control problems

We study a residual–based a posteriori error estimate for the solution of Dirichlet boundary control problem governed by a convection diffusion equation on a two dimensional convex polygonal domain, using the local discontinuous Galerkin (LDG) method with upwinding for the convection term. With the usage of LDG method, the control variable naturally exists in the variational form due to its mixed finite element structure. We also demonstrate the application of our a posteriori error estimator for the adaptive solution of these optimal control problems.

$L^\infty$ Norm Error Estimates for HDG Methods Applied to the Poisson Equation with an Application to the Dirichlet Boundary Control Problem

We prove quasi-optimal $L^\infty$ norm error estimates (up to logarithmic factors) for the solution of Poisson's problem in two dimensional space by the standard hybridizable discontinuous Galerkin (HDG) method. Although such estimates are available for conforming and mixed finite element methods, this is the first proof for HDG. The method of proof is motivated by known $L^\infty$ norm estimates for mixed finite elements. We show two applications: the first is to prove optimal convergence rates for boundary flux estimates, and the second is to prove that numerically observed convergence rates for the solution of a Dirichlet boundary control problem are to be expected theoretically. Numerical examples show that the predicted rates are seen in practice.

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with anL2penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.

A Finite Element Method for Elliptic Dirichlet Boundary Control Problems

Abstract We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in H 1 / 2 ⁢ ( Γ ) {H^{1/2}(\Gamma)} . To avoid computing the latter norm numerically, we realize it using the H 1 ⁢ ( Ω ) {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 {H^{1}} and L 2 {L_{2}} norm are proven. We also consider and analyze the case of control constrained problems.

Goal–oriented a posteriori error estimation for Dirichlet boundary control problems

We study goal-oriented a posteriori error estimates for the numerical approximation of Dirichlet boundary control problem governed by a convection diffusion equation with pointwise control constraints on a two dimensional convex polygonal domain. The local discontinuous Galerkin method is used as a discretization technique since the control variable is involved in a variational form in a natural sense. We derive primal–dual weighted error estimates for the objective functional with an error term representing the mismatch in the complementary system due to the discretization. Numerical examples are presented to illustrate the performance of the proposed estimator.

Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs

We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems governed by elliptic PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. We propose an HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control. Specifically, under certain assumptions, for a 2D convex polygonal domain we show the control converges at a superlinear rate. We present 2D and 3D numerical experiments to demonstrate our theoretical results.