A Hybridized Discontinuous Galerkin Method for A Linear Degenerate Elliptic Equation Arising from Two-Phase Mixtures
A Hybridized Discontinuous Galerkin Method for A Linear Degenerate Elliptic Equation Arising from Two-Phase Mixtures
复制标题
DOI：
10.1016/j.cma.2019.03.018
复制
发表时间：
2018-08
期刊：
影响因子：
--
通讯作者：
Shinhoo Kang;T. Bui-Thanh;T. Arbogast
中科院分区：
文献类型：
--
作者：
Shinhoo Kang;T. Bui-Thanh;T. Arbogast
我们发展了一种高阶杂交间断Galerkin（HDG）方法，用于求解由地幔对流或冰川动力学两相混合引起的线性退化椭圆型方程。我们表明，建议的HDG方法是适定的，通过使用能量的方法。我们推导出先验误差估计的方法在二维和三维的单纯网格。误差分析表明，对于非退化问题，其收敛速度对标度压力和标度速度都是最优的，而对于退化问题，其收敛速度是次优的。几个数值结果证实了理论估计。我们还通过后处理增强HDG解决方案。对于非退化情形和远离退化的退化情形，都观察到（k+ 2）和（k+32）的超收敛速度.退化问题的低正则性的解决方案也进行了研究，数值结果表明，高阶方法是有益的精度。
We develop a high-order hybridized discontinuous Galerkin (HDG) method for a linear degenerate elliptic equation arising from a two-phase mixture of mantle convection or glacier dynamics. We show that the proposed HDG method is well-posed by using an energy approach. We derive a priori error estimates for the method on simplicial meshes in both two-and three-dimensions. The error analysis shows that the convergence rates are optimal for both the scaled pressure and the scaled velocity for non-degenerate problems and are sub-optimal by half order for degenerate ones. Several numerical results are presented to confirm the theoretical estimates. We also enhance the HDG solutions by post-processing. The superconvergence rates of (k+ 2) and (k+ 3 2) are observed for both a non-degenerate case and a degenerate case away from the degeneracy. Degenerate problems with low regularity solutions are also studied, and numerical results show that high-order methods are beneficial in terms of accuracy.