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Journal of Scientific Computing

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26-02-2018

A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs

Authors: Weiwei Hu, Jiguang Shen, John R. Singler, Yangwen Zhang, Xiaobo Zheng

Published in: Journal of Scientific Computing | Issue 3/2018

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Abstract

We consider a distributed optimal control problem governed by an elliptic convection diffusion PDE, and propose a hybridizable discontinuous Galerkin method to approximate the solution. We use polynomials of degree \(k+1\) to approximate the state and dual state, and polynomials of degree \(k \ge 0\) to approximate their fluxes. Moreover, we use polynomials of degree k to approximate the numerical traces of the state and dual state on the faces, which are the only globally coupled unknowns. We prove optimal a priori error estimates for all variables when \( k \ge 0 \). Furthermore, from the point of view of the number of degrees of freedom of the globally coupled unknowns, this method achieves superconvergence for the state, dual state, and control when \(k\ge 1\). We illustrate our convergence results with numerical experiments.

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Title: A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs
Authors: Weiwei Hu
Jiguang Shen
John R. Singler
Yangwen Zhang
Xiaobo Zheng
Publication date: 26-02-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0668-z

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