一种用于两相混合物中线性退化椭圆方程的混合不连续伽辽金方法
A Hybridized Discontinuous Galerkin Method for A Linear Degenerate Elliptic Equation Arising from Two-Phase Mixtures
AI总结：
本文提出了一种高阶混合不连续伽辽金方法，用于解决源于 mantle convection 或 glacier dynamics 的线性退化椭圆方程，通过能量方法证明了方法的良定性，并推导了在二维和三维 simplicial 网格上的 a priori 错误估计，展示了非退化和退化问题的收敛率，还通过数值实验验证了理论估计，并通过后处理提升了 HDG 解的超收敛率。
摘要：
我们为源于 mantle convection 或 glacier dynamics 的线性退化椭圆方程开发了一种高阶混合不连续伽辽金 (HDG) 方法。我们通过能量方法证明了所提出的 HDG 方法的良定性。我们推导了所提出的 HDG 方法在二维和三维 simplicial 网格上的 a priori 错误估计。误差分析表明，对于非退化问题，缩放压力和缩放速度的收敛率是最佳的，而对于退化问题，收敛率则降低了一半的阶数。若干数值结果被呈现以验证理论估计。我们还通过后处理增强了 HDG 解。对于非退化情况和远离退化情况的退化情况，均观察到超收敛率分别为 (k+2) 和 (k+3/2)。还研究了具有低正则性解的退化问题，数值结果表明高阶方法在精度方面具有优势。
英文摘要：
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 ${\it a priori}$ error estimates for the proposed HDG 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 $(k+\frac{3}{2})$ 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.
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