A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs
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 ≥ 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 ≥ 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 ≥ 1. We illustrate our convergence results with numerical experiments.
W. Hu et al., "A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs," Journal of Scientific Computing, vol. 76, no. 3, pp. 1436-1457, Springer, Sep 2018.
The definitive version is available at https://doi.org/10.1007/s10915-018-0668-z
Mathematics and Statistics
Keywords and Phrases
Convection diffusion equation; Distributed optimal control; Error analysis; Hybridizable discontinuous Galerkin method; Superconvergence
International Standard Serial Number (ISSN)
Article - Journal
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01 Sep 2018