A Legendre–Galerkin Spectral Method for Optimal Control Problems Governed by Stokes Equations

Chen, Yanping; Huang, Fenglin; Yi, Nianyu; Liu, Wenbin

doi:10.1137/080726057

Cited by 43 publications

(21 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now, we introduce the co-state elliptic equation By modifying the proofs of Lemma 2.1 in [7] and Lemma 3.2 in [12], using the regularity argument of elliptic problems in [2,3], we can get that y,z e H^ 640 Y. Chen and T. Hou In this paper, we shall employ duality respect to H^n) in place of H¿(n); i.e., if (/?…”

Section: Mixed Methods For Optimal Control Problemsmentioning

confidence: 99%

Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

Chen¹

2013

NMTMA

View full text Add to dashboard Cite

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive L°°-and H"^-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

show abstract

Section: Mixed Methods For Optimal Control Problemsmentioning

confidence: 99%

Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

Chen¹

2013

NMTMA

View full text Add to dashboard Cite

show abstract

“…In recent years, we have investigated a mixed finite element method for optimal control problems, whose objective functional contains both flux and state variables. We have carried out carefully some research on superconvergence properties, a priori error estimates, and a posteriori error estimates to validate our mixed finite element methods for optimal control problems and high-order accuracy, spectral methods for some flow control problems; see [22][23][24][25]. Thus it is worth to investigate mixed finite element methods for the control problems involving flux control.…”

Section: Introductionmentioning

confidence: 97%

A mixed multiscale finite element method for convex optimal control problems with oscillating coefficients

Chen

Huang

Liu

et al. 2015

Computers & Mathematics with Applications

Self Cite

View full text Add to dashboard Cite

Available online xxxx Keywords:Multiscale control problem Mixed finite element Homogenization A prior error estimates a b s t r a c tWe study numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients. Since the objective functional contains flux, we approximate the problems using the mixed finite element methods. We first analyze the standard mixed finite element approximation schemes. Then, motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media, we use a multiscale mixed finite element method to solve the state equations. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. The analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic, which allows us to use homogenization theory to obtain the asymptotic structure of the solutions, although the numerical schemes are designed for the general case.

show abstract

“…Until now, finite element approximation of optimal control problems has been studied extensively, see, for example, [8,15,16,23,24,28]. In recent years, spectral method has been used to approximate the optimal control problems (see e.g., [11,12]). However most of these work focus on control-constrained problems.…”

Section: Introductionmentioning

confidence: 99%

Galerkin Spectral Approximation of Elliptic Optimal Control Problems with $$H^1$$ H 1 -Norm State Constraint

Chen

Huang

2015

J Sci Comput

Self Cite

View full text Add to dashboard Cite

In this paper, we study an elliptic optimal control problem with H 1 -norm state constraint. The control problem is approximated by the Galerkin spectral method, which can provide high-order accuracy and fast convergence rate. The optimality conditions and a priori error estimates are presented. A reliable a posteriori error estimator is investigated, which is helpful for developing adaptive strategy in the spectral method. Some numerical tests confirm the error estimates and illustrate the performance of the indicator.

show abstract

A Legendre–Galerkin Spectral Method for Optimal Control Problems Governed by Stokes Equations

Cited by 43 publications

References 17 publications

Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

A mixed multiscale finite element method for convex optimal control problems with oscillating coefficients

Galerkin Spectral Approximation of Elliptic Optimal Control Problems with $$H^1$$ H 1 -Norm State Constraint

Contact Info

Product

Resources

About