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：我们将高阶混合有限元离散化技术及其相关的预处理迭代求解器应用于二维情况下的变量埃丁顿因子（VEF）方程。混合有限元 VEF 离散化与高阶不连续 Galerkin（DG）离散化的离散正交传输方程相耦合，形成与高阶（曲线）网格兼容的有效线性传输算法。VEF 和传输离散化的组合受到劳伦斯·利弗莫尔国家实验室在流体动力学计算中使用高阶混合有限元方法的启发。由于 VEF 方程的数学结构，标准的 Raviart-Thomas（RT）混合有限元不能用于近似 VEF 方程中的矢量变量。因此，我们研究了三种替代方案，基于将连续有限元用于每个矢量分量，采用 DG 类技术的非协调 RT 方法以及混合 RT 方法。我们提供的数值结果证明了高阶精度、曲线网格的兼容性以及迭代求解耦合的传输-VEF 系统和用于求解离散 VEF 方程的预处理线性求解器的稳健和有效的收敛性。