We consider anisotropic heat flow with extreme anisotropy, as arises in magnetized plasmas for fusion applications. Such problems pose significant challenges in both obtaining an accurate approximation as well in the construction of an efficient solver. In both cases, the underlying difficulty is in forming an accurate approximation of temperature fields that follow the direction of complex, non-grid-aligned magnetic fields. In this work, we construct a highly accurate coarse grid approximation using spectral multiscale basis functions based on local anisotropic normalized Laplacians. We show that the local generalized spectral problems yield local modes that align with magnetic fields, and provide an excellent coarse-grid approximation of the problem. We then utilize this spectral coarse space as an approximation in itself, and as the coarse-grid in a two-level spectral preconditioner. Numerical results are presented for several magnetic field distributions and anisotropy ratios up to $10^{12}$, showing highly accurate results with a large system size reduction, and two-grid preconditioning that converges in $O(1)$ iterations, independent of anisotropy.

翻译：我们研究极端各向异性条件下的热传导问题，该问题在聚变应用的磁化等离子体中普遍存在。此类问题在获取精确近似与构建高效求解器两方面均面临重大挑战。两者的核心困难在于如何对沿复杂非网格对齐磁场方向分布的温度场形成精确近似。本工作基于局部各向异性归一化拉普拉斯算子，利用谱多尺度基函数构建了高精度粗网格近似。我们证明局部广义谱问题产生的局部模态能与磁场方向对齐，并为原问题提供了优异的粗网格近似。随后，我们将该谱粗空间同时用作独立近似与双层谱预条件子的粗网格层。针对多种磁场分布及各向异性比高达$10^{12}$的数值实验表明：该方法在实现大幅系统规模缩减的同时获得高精度结果，且双网格预条件子可在$O(1)$迭代次数内收敛，其收敛性与各向异性程度无关。