SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 720-745, January 2021.
We prove quasi-optimal $L^\infty$ norm error estimates (up to logarithmic factors) for the solution of Poisson's problem in two dimensional space by the standard hybridizable discontinuous Galerkin (HDG) method. Although such estimates are available for conforming and mixed finite element methods, this is the first proof for HDG. The method of proof is motivated by known $L^\infty$ norm estimates for mixed finite elements. We show two applications: the first is to prove optimal convergence rates for boundary flux estimates, and the second is to prove that numerically observed convergence rates for the solution of a Dirichlet boundary control problem are to be expected theoretically. Numerical examples show that the predicted rates are seen in practice.

中文翻译：

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用于Poisson方程的HDG方法的$ L ^ \ infty $范数误差估计及其在Dirichlet边界控制问题中的应用

SIAM数值分析学报，第59卷，第2期，第720-745页，2021年1月。
我们通过标准的可杂交不连续伽勒金（HDG）方法证明了二维空间中泊松问题解的拟最优$ L ^ \ infty $范数误差估计（高达对数因子）。尽管这样的估计可用于一致性和混合有限元方法，但这是HDG的第一个证明。证明方法是由混合有限元的已知$ L ^ \ infty $范数估计所激发的。我们展示了两个应用程序：第一个是证明边界通量估计的最优收敛速度，第二个是证明在理论上期望对Dirichlet边界控制问题的解进行数值观测的收敛速度。数值示例表明，在实际中可以看到预测的速率。