一种用于椭圆PDE Dirichlet边界控制的超收敛混合不连续Galerkin方法
A Superconvergent Hybridizable Discontinuous Galerkin Method for Dirichlet Boundary Control of Elliptic PDEs
AI总结：
本文提出了一种用于椭圆PDE Dirichlet边界控制问题的混合不连续Galerkin方法，获得了最优先验误差估计，并在二维和三维数值实验中验证了理论结果。
AI中文摘要：
我们开始研究用于近似由椭圆PDEs支配的Dirichlet边界控制问题的混合不连续Galerkin（HDG）方法。这些问题可能涉及非典型变分公式，且在多面体域上解的正则性较低。这些问题可能对数值方法及其相关数值分析构成挑战。我们为泊松方程的Dirichlet边界控制问题提出了一种HDG方法，并获得了控制的最优先验误差估计。具体而言，在某些假设下，对于二维凸多边形域，我们证明控制以超线性速率收敛。我们展示了二维和三维数值实验以验证我们的理论结果。
英文摘要：
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.
