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A projective hybridizable discontinuous Galerkin mixed method for second-order diffusion problems

机译：二阶扩散问题的射影可杂交不连续Galerkin混合方法

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In this paper, we present a hybridizable discontinuous Galerkin (HDG) mixed method for second-order diffusion problems using a projective stabilization function and broken Raviart-Thomas functions to approximate the dual variable. The proposed HDG mixed method is inspired by the primal HDG scheme with reduced stabilization suggested by Lehrenfeld and Schoberl in 2010, and the standard hybridized version of the Raviart-Thomas (H-RT) method. Indeed, we use the broken Raviart-Thomas space of degree k > 0 for the flux, a piecewise polynomial of degree k + 1 for the potential, and a piecewise polynomial of degree k for its numerical trace. This unconventional polynomial combination is made possible by the projective Lehrenfeld-Schoberl (LS) stabilization function. Its introduction and the use of Raviart-Thomas spaces will have beneficial effects: no postprocessing is required to improve the accuracy of the potential u h , and a straightforward flux reconstruction is sufficient to obtain a H(div)-conforming flux variable. The convergence and accuracy of our method are investigated through numerical experiments in two-dimensional space by using h and p refinement strategies. An optimal convergence order (k + 1) for the H(div)-conforming flux and superconvergence (k + 2) for the potential is observed. ComParative tests with the classical H-RT and the well-known hybridizable local discontinuous Galerkin (H-LDG) mixed methods are also performed and exposed in terms of CPU time. (C) 2019 Elsevier Inc. All rights reserved.
机译：在本文中，我们提出了一种利用射影稳定函数和破碎的Raviart-Thomas函数近似二元变量的二阶扩散问题的可混合不连续Galerkin（HDG）混合方法。拟议的HDG混合方法的灵感来自于Lehrenfeld和Schoberl在2010年提出的具有降低的稳定性的原始HDG方案，以及Raviart-Thomas（H-RT）方法的标准杂交版本。实际上，我们对通量使用了k> 0的破裂Raviart-Thomas空间，对势使用了k + 1的分段多项式，对它的数值迹线使用了k的分段多项式。投影的Lehrenfeld-Schoberl（LS）稳定函数使这种非常规的多项式组合成为可能。它的引入和Raviart-Thomas空间的使用将产生有益的效果：不需要后处理来提高势u h的精度，直接进行磁通重构就足以获得符合H（div）的磁通变量。通过使用h和p细化策略在二维空间中进行数值实验，研究了我们方法的收敛性和准确性。观察到符合H（div）的通量的最佳收敛阶（k +1）和势能的超收敛（k + 2）。还使用经典H-RT和众所周知的可混合局部不连续Galerkin（H-LDG）混合方法进行了比较测试，并根据CPU时间进行了测试。 （C）2019 Elsevier Inc.保留所有权利。

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