We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Discrete Ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory. Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations.

翻译：本文将高阶混合有限元离散技术及其相关预处理迭代求解器应用于二维空间中的变爱丁顿因子（VEF）方程。混合有限元VEF离散格式与离散纵标输运方程的高阶间断伽辽金（DG）离散格式耦合，形成与高阶（弯曲）网格兼容的高效线性输运算法。这种VEF与输运离散格式的组合源于劳伦斯利弗莫尔国家实验室在流体力学计算中对高阶混合有限元方法的应用需求。由于VEF方程的数学结构，标准Raviart-Thomas（RT）混合有限元无法用于逼近VEF方程中的向量变量。为此，我们研究了三种替代方案：基于每个向量分量的连续有限元方法、采用类DG技术的非协调RT方法以及混合化RT方法。数值结果表明，该方法具有高阶精度、与弯曲网格的兼容性，并在迭代求解耦合输运-VEF系统及用于离散VEF方程反演的预处理线性求解器中展现出稳健高效的收敛性能。