A New HDG Method for Dirichlet Boundary Control of Convection Diffusion PDEs II: Low Regularity


In the first part of this work, we analyzed an unconstrained Dirichlet boundary control problem for an elliptic convection diffusion PDE and proposed a new hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For the case of a 2D convex polygonal domain, we also proved an optimal superlinear convergence rate for the control under certain assumptions on the domain and on the target state. In this work, we revisit the convergence analysis without these assumptions; in this case, the solution can have low regularity, and we use a different analysis approach. We again prove an optimal convergence rate for the control and present numerical results to illustrate the convergence theory.


Mathematics and Statistics


The work of the first author was supported by the National Natural Science Foundation of China under grants 11671391 and 91530204. The work of the second author was partially supported by the DIG and FY 2018 ASR+1 Program at Oklahoma State University. The work of the third author was supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P. The work of the fourth and sixth authors was supported in part by National Science Foundation grant DMS-1217122. The fourth and sixth authors also acknowledge the support of the Institute for Mathematics and its Applications at the University of Minnesota in funding research visits, during which some of this work was completed. The fifth author acknowledges the support of Missouri University of Science and Technology, which hosted him as a visiting scholar; some of this work was completed during his research visit.

Keywords and Phrases

Diffusion in liquids; Error analysis; Heat convection; Convection diffusion; Convergence analysis; Convex polygonal domain; Dirichlet boundary controls; Discontinuous galerkin; Discontinuous Galerkin methods; Low regularity; Superlinear convergence rate; Galerkin methods; Hybridizable discontinuous Galerkin method (HDG)

International Standard Serial Number (ISSN)

0036-1429; 1095-7170

Document Type

Article - Journal

Document Version


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Publication Date

01 Jul 2018