We first propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a convection dominated Dirichlet boundary control problem without constraints. Dirichlet boundary control problems and convection dominated problems are each very challenging numerically due to solutions with low regularity and sharp layers, respectively. Although there are some numerical analysis works in the literature on diffusion dominated convection diffusion Dirichlet boundary control problems, we are not aware of any existing numerical analysis works for convection dominated boundary control problems. Moreover, the existing numerical analysis techniques for convection dominated PDEs are not directly applicable for the Dirichlet boundary control problem because of the low regularity solutions. In this work, we obtain an optimal a priori error estimate for the control under some conditions on the domain and the desired state. We also present some numerical experiments to illustrate the performance of the HDG method for convection dominated Dirichlet boundary control problems.
G. Chen et al., "An HDG Method for Dirichlet Boundary Control of Convection Dominated Diffusion PDEs," SIAM Journal on Numerical Analysis, vol. 57, no. 4, pp. 1919 - 1946, Society for Industrial and Applied Mathematics (SIAM), Aug 2019.
The definitive version is available at https://doi.org/10.1137/18M1208708
Mathematics and Statistics
Center for High Performance Computing Research
Keywords and Phrases
Convection dominated diffusion PDEs; Dirichlet boundary control; Error analysis; HDG method; Hybridizable discontinuous Galerkin method; Low regularity
International Standard Serial Number (ISSN)
Article - Journal
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13 Aug 2019