A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations

Akman, Tuğba; Yücel, Hamdullah; Karasözen, Bülent

doi:10.1007/s10589-013-9601-4

Cited by 14 publications

(14 citation statements)

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“…For details, we refer the reader to the study [19] where a priori error estimates for SIPG discretization combined with backward Euler is provided and the quadratic convergence rate in space is achieved. For time-discretization, we use Crank-Nicolson method, which is known to be stable and second order convergent.…”

Section: Problem Discretizationmentioning

confidence: 99%

Local improvements to reduced-order approximations of optimal control problems governed by diffusion–convection–reaction equation

Akman

2015

Computers & Mathematics with Applications

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We consider the optimal control problem governed by diffusion convection reaction equation without control constraints. The proper orthogonal decomposition(POD) method is used to reduce the dimension of the problem. The POD method may be lack of accuracy if the POD basis depending on a set of parameters is used to approximate the problem depending on a different set of parameters. We are interested in the perturbation of diffusion term. To increase the accuracy and robustness of the basis, we compute three bases additional to the baseline POD. The first two of them use the sensitivity information to extrapolate and expand the POD basis. The other one is based on the subspace angle interpolation method. We compare these different bases in terms of accuracy and complexity and investigate the advantages and main drawbacks of them.

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Section: Problem Discretizationmentioning

confidence: 99%

Local improvements to reduced-order approximations of optimal control problems governed by diffusion–convection–reaction equation

Akman

2015

Computers & Mathematics with Applications

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“…The dG methods have several advantages compared to other numerical techniques such as finite volume and finite element methods; the trial and test spaces can be easily constructed, inhomogeneous boundary conditions and curved boundaries can be handled easily. The dG methods were successfully applied to linear steady state, time dependent and semi-linear optimal control problems with convectiondiffusion-reaction equations [11,12,13]; to the semi-linear steady state OCPs [14]. There are two approaches for solving OCPs with PDE constraints.…”

Section: Introductionmentioning

confidence: 99%

Reduced order optimal control of the convective FitzHugh–Nagumo equations

Karasözen

Uzunca

Küçükseyhan

2020

Computers & Mathematics with Applications

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We compare three different model order reduction techniques with Galerkin projection: the proper orthogonal decomposition (POD), POD-DEIM (discrete empirical interpolation) and POD-DMD (dynamic mode decomposition) for solving optimal control problems governed by the convective FitzHugh-Nagumo (FHN) equation. The convective FHN equation consists of the semilinear activator and the linear inhibitor equation, modelling blood coagulation in moving excitable media. The POD and POD-DEIM reduced optimal control problems are nonconvex due to the nonlinear activator equation. DMD is an equation-free, datadriven method which extracts dynamically relevant information content without explicitly knowing the dynamical operator. We use DMD as an alternative method to DEIM in order to approximate the nonlinear term in the convective FHN equation. Applying the POD-DMD Galerkin projection gives rise to a linear system of equations for the activator, and the optimal control problem becomes convex. We compare the accuracy and CPU times of three reduced order methods(ROM) with respect to the full order discontinuous Galerkin finite element solutions for convection dominated wave type solutions with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest.

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“…Several well-established techniques have been proposed to enhance stability and accuracy of the optimal control problems governed by the steady convection diffusion equation, e.g., the streamline upwind/Petrov Galerkin (SUPG) finite element method [11], the local projection stabilization [5], the edge stabilization [27,51], and discontinuous Galerkin methods [32,52,53,54,55]. However, only few papers are published so far for unsteady optimal control problems governed by convection diffusion equations, e.g., the characteristic finite element method [14,15], the streamline upwind/Petrov Galerkin (SUPG) finite element method [30], the local discontinuous Galerkin (LDG) method [57], the nonsymmetric interior penalty Galerkin (NIPG) method [45], and the symmetric interior penalty Galerkin (SIPG) method [2].…”

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confidence: 99%

“…Note that, given the reduced regularity of the control functions imposed by the control constraints, natural candidates for discretizing the control space would be the piecewise constant finite elements. Nevertheless, we use piecewise linear polynomials as is often done in the literature [2,5,43], resulting in the optimal convergence rate h 3/2 U in space for the control. We can now give the DG discretizations of the state equation (22) in space for a fixed control u.…”

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Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations

Yücel

Stoll

Benner

2018

etna

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In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. A symmetric interior penalty Galerkin method in space and a backward Euler method in time are applied in order to discretize the optimization problem. Numerical results are presented, which illustrate the performance of the proposed error estimators.

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A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations

Cited by 14 publications

References 20 publications

Local improvements to reduced-order approximations of optimal control problems governed by diffusion–convection–reaction equation

Local improvements to reduced-order approximations of optimal control problems governed by diffusion–convection–reaction equation

Reduced order optimal control of the convective FitzHugh–Nagumo equations

Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations

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