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

Cagnur Corekli; Cagnur Corekli

doi:10.3934/math.2022375

AIMS Mathematics

2022, Volume 7, Issue 4: 6711-6742. doi: 10.3934/math.2022375

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Research article

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

Cagnur Corekli ^,

Department of Finance and Banking, Recep Tayyip Erdogan University, Rize, 53700, Turkey
This work is part of the author's Ph.D. thesis, prepared at the University of Connecticut, CT, USA, 2016

Received: 19 July 2021 Revised: 21 December 2021 Accepted: 12 January 2022 Published: 25 January 2022
MSC : 33F05, 65K10, 65M99, 65N30, 65N99

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.
- finite elements,
- discontinuous Galerkin methods,
- SIPG method,
- Dirichlet boundary,
- transposition method,
- optimal control,
- stabilized methods
Citation: Cagnur Corekli. The SIPG method of Dirichlet boundary optimal control problems with weakly imposed boundary conditions[J]. AIMS Mathematics, 2022, 7(4): 6711-6742. doi: 10.3934/math.2022375

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Abstract

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.

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