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A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs

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Abstract

We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems governed by elliptic PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. We propose an HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control. Specifically, under certain assumptions, for a 2D convex polygonal domain we show the control converges at a superlinear rate. We present 2D and 3D numerical experiments to demonstrate our theoretical results.

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Acknowledgements

The authors thank the referees for their comments, which helped improve the manuscript. W. Hu was supported in part by a postdoctoral fellowship for the annual program on Control Theory and its Applications at the Institute for Mathematics and its Applications (IMA) at the University of Minnesota, and also the DIG and FY 2018 ASR+1 Program at Oklahoma State University. J. Singler and Y. Zhang were supported in part by National Science Foundation grant DMS-1217122. J. Singler and Y. Zhang thank the IMA for funding research visits, during which some of this work was completed. X. Zheng thanks Missouri University of Science and Technology for hosting him as a visiting scholar; some of this work was completed during his research visit. The authors thank Bernardo Cockburn for many valuable conversations.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, Stillwater, OK, USA

    Weiwei Hu

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, USA

    Jiguang Shen

  3. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

    John R. Singler & Yangwen Zhang

  4. College of Mathematics, Sichuan University, Chengdu, China

    Xiaobo Zheng

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  1. Weiwei Hu
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  2. Jiguang Shen
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  3. John R. Singler
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By simple algebraic operations in Eq. (17), we obtain the following formulas for the matrices \( G_1 \), \( G_2 \), \( H_1 \), and \( H_2 \) in (18):

$$\begin{aligned} G_1&= B_1^{-1}B_2(B_4+B_2^TB_1^{-1}B_2)^{-1}(B_5+B_2^TB_1^{-1}B_3)-B_1^{-1}B_3,\\ G_2&= -\,(B_4+B_2^TB_1^{-1}B_2)^{-1}(B_5+B_2^TB_1^{-1}B_3),\\ H_1&= -\,B_1^{-1}B_2(B_4+B_2^TB_1^{-1}B_2)^{-1},\\ H_2&= (B_4+B_2^TB_1^{-1}B_2)^{-1}. \end{aligned}$$

In general, this process is impractical; however, for the HDG method described in this work, these matrices can be easily computed. This is one of the advantages of the HDG method. We briefly describe this process below.

Since the spaces \( \varvec{V}_h \) and \( W_h \) consist of discontinuous polynomials, some of the system matrices are block diagonal and each block is small and symmetric positive definite. Let us call a matrix of this form a SSPD block diagonal matrix. The inverse of a SSPD block diagonal matrix is another SSPD block diagonal matrix, and the inverse can be easily constructed by computing the inverse of each small block. Furthermore, the inverse of each small block can be computed independently; and therefore computing the inverse can be easily done in parallel.

It can be checked that \( B_1 \) is a SSPD block diagonal matrix, and therefore \( B_1^{-1} \) is easily computed and is also a SSPD block diagonal matrix. Therefore, the matrices \( G_1 \), \( G_2 \), \( H_1 \), and \( H_2 \) are easily computed if \( B_4 + B_2^T B_1^{-1} B_2 \) is also easily inverted. We show below that this is the case.

First, it can be checked that \( B_2 \) is block diagonal with small blocks, but the blocks are not symmetric or definite. This implies \(B_2^T B_1^{-1} B_2\) is block diagonal with small nonnegative definite blocks. Next, \(B_4 = \begin{bmatrix} A_5&0\\ -A_4&A_5 \end{bmatrix}\), where \(A_4\) and \(A_5\) are both SSPD block diagonal. Due to the structure of \( B_1 \) and \( B_2 \), the matrix \(B_2^TB_1^{-1}B_2 + B_4\) has the form \(\begin{bmatrix} C_1&0\\ -A_4&C_2 \end{bmatrix}, \) where \( C_1 \) and \( C_2 \) are SSPD block diagonal. The inverse can be easily computed using the formula

$$\begin{aligned}\begin{bmatrix} C_1&0\\ -A_4&C_2 \end{bmatrix}^{-1} = \begin{bmatrix} C_1^{-1}&0\\ C_2^{-1} A_4 C_1^{-1}&C_2^{-1} \end{bmatrix}. \end{aligned}$$

Furthermore, \( C_1^{-1} \), \(C_2^{-1}\) and \( C_2^{-1} A_4 C_1^{-1} \) are both SSPD block diagonal.

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Hu, W., Shen, J., Singler, J.R. et al. A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs. Numer. Math. 144, 375–411 (2020). https://doi.org/10.1007/s00211-019-01090-2

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