Extended finite element methods for optimal control problems governed by Poisson equation in non-convex domains

Wang, Tao; Yang, Chaochao; Xie, Xiaoping

doi:10.1007/s11425-018-9487-4

Cited by 5 publications

(2 citation statements)

References 46 publications

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“…The optimal control problem (1) subject to an elliptic or heat equation is a classic problem, which has been thoroughly studied both in theoretical and numerical aspects; see, e.g. [9,20,38] for theoretical analysis and [4,7,15,18,21,40,41] for finite element analysis. In general, there are mainly two discretization concepts for this problem: direct and variational discretizations.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

Wang¹,

Li²,

Xie³

2020

Preprint

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This paper is devoted to the numerical analysis of a control constrained distributed optimal control problem subject to a time fractional diffusion equation with non-smooth initial data. The solutions of state and costate are decomposed into singular and regular parts, and some growth estimates are obtained for the singular parts. Following the variational discretization concept, a full discretization is applied to the corresponding state and co-state equations by using linear conforming finite element method in space and piecewise constant discontinuous Galerkin method in time. By the growth estimates, error estimates are derived with nonsmooth initial data. In particular, graded temporal grids are used to obtain the first-order temporal accuracy. Finally, numerical experiments are performed to verify the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

Wang¹,

Li²,

Xie³

2020

Preprint

Self Cite

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“…Nonetheless, the effectiveness of these methods for achieving optimal control in PDEs has not been suitably assessed. The necessity of optimal control in PDEs is ubiquitous throughout applied sciences and engineering and extensive literature on the analysis of such problems is readily available, see, for example, [18,19,21,34,46,[61][62][63]65] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Random geometries for optimal control PDE problems based on fictitious domain FEMS and cut elements

Aretaki¹,

Karatzas²

2020

Preprint

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This work investigates an elliptic optimal control problem defined on uncertain domains and discretized by a fictitious domain finite element method and cut elements. Key ingredients of the study are to manage cases considering the usually computationally "forbidden" combination of poorly conditioned equation system matrices due to challenging geometries, optimal control searches with iterative methods, slow convergence to system solutions on deterministic and non-deterministic level, and expensive remeshing due to geometrical changes. We overcome all these difficulties, utilizing the advantages of proper preconditioners adapted to unfitted mesh methods, improved types of Monte Carlo methods, and mainly employing the advantages of embedded FEMs, based on a fixed background mesh computed once even if geometrical changes are taking place. The sensitivity of the control problem is introduced in terms of random domains, employing a Quasi-Monte Carlo method. The variational discretization concept is adopted, optimal error estimates for the state, adjoint state and control are derived that confirm the efficiency of the cut finite element method in challenging geometries. The performance of a multigrid scheme especially developed for unfitted finite element discretizations adapted to the optimal control problem is also tested. Some fundamental preconditioners are applied to the arising sparse linear systems coming from the discretization of the state and adjoint state variational forms in the spatial domain. The corresponding convergence rates along with the quality of the prescribed preconditioners are verified by numerical examples. Further, a practical error estimate for the Quasi-Monte Carlo method is derived, showing that it yields the theoretically predicted convergence rate.

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An Interface/Boundary-Unfitted EXtended HDG Method for Linear Elasticity Problems

2023

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Extended finite element methods for optimal control problems governed by Poisson equation in non-convex domains

Cited by 5 publications

References 46 publications

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

Random geometries for optimal control PDE problems based on fictitious domain FEMS and cut elements

An Interface/Boundary-Unfitted EXtended HDG Method for Linear Elasticity Problems

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