We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. We simultaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators. We pose the problem in a mixed formulation, introducing the heat flux as an auxiliary variable and discretizing the temperature and auxiliary fields in a discontinuous Galerkin space. The resulting block matrix system is then reordered and solved using an approach in which two advection operators are inverted using AMG solvers based on approximate ideal restriction (AIR), which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are non-singular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines. We demonstrate the proposed discretization's superior accuracy over other discretizations of anisotropic heat flux, achieving error $1000\times$ smaller for anisotropy ratio of $10^9$, while also demonstrating fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.

翻译：我们提出了一种针对各向异性热通量方程的新型求解器技术，旨在解决磁约束聚变等离子体中存在的高度各向异性问题。此类问题面临两大挑战：(i) 离散化精度与(ii) 高效隐式线性求解器。我们通过构建一种具有优异精度特性的新有限元离散化，并针对基于对流算子设计的代数多重网格（AMG）新型求解方法进行定制，同时应对了这些挑战。我们采用混合公式描述问题，将热通量作为辅助变量引入，并在间断伽辽金空间中对温度场和辅助场进行离散化。由此产生的块矩阵系统经过重排序后，采用基于近似理想限制（AIR）的AMG求解器对两个对流算子进行求逆，该方法对于对流问题的迎风间断伽辽金离散化特别高效。为确保对流算子非奇异，本文仅考虑开放（非循环）磁力线。我们证明了所提离散化在各项异性热通量方面优于其他离散化方法：当各向异性比为10⁹时，误差减小了1000倍，同时该迭代求解器在高度各向异性区域（其他基于扩散的AMG方法失效）展现出快速收敛性。