Highly accurate simulation of plasma transport is needed for the successful design and operation of magnetically confined fusion reactors. Unfortunately, the extreme anisotropy present in magnetized plasmas results in thin boundary layers that are expensive to resolve. This work investigates how mesh refinement strategies might reduce that expense to allow for more efficient simulation. It is first verified that higher order discretization only realizes the proper rate of convergence once the mesh resolves the thin boundary layer, motivating the focusing of refinement on the boundary layer. Three mesh refinement strategies are investigated: one that focuses the refinement across the layer by using rectangular elements with a ratio equal to the boundary layer width, one that allows for exponential growth in mesh spacing away from the layer, and one adaptive strategy utilizing the established Zienkiewicz and Zhu error estimator. Across 4 two-dimensional test cases with high anisotropy, the adaptive mesh refinement strategy consistently achieves the same accuracy as uniform refinement using orders of magnitude less degrees of freedom. In the test case where the magnetic field is aligned with the mesh, the other refinement strategies also show substantial improvement in efficiency. This work also includes a discussion generalizing the results to larger magnetic anisotropy ratios and to three-dimensional problems. It is shown that isotropic mesh refinement requires degrees of freedom on the order of either the layer width (2D) or the square of the layer width (3D), whereas anisotropic refinement requires a number on the order of the log of layer width for all dimensions. It is also shown that the number of conjugate gradient iterations scales as a power of layer width when preconditioned with algebraic multigrid, whereas the number is independent of layer width when preconditioned with ILU.

翻译：磁约束聚变反应堆的成功设计与运行需要对等离子体输运进行高精度模拟。然而，磁化等离子体中存在的极端各向异性会导致薄边界层的产生，从而增加求解成本。本研究探讨了网格细化策略如何降低这一成本以实现更高效的模拟。首先验证了高阶离散化只有在网格解析薄边界层时才能达到正确的收敛速率，这促使将细化重点集中于边界层。研究了三种网格细化策略：一种通过使用宽高比等于边界层宽度的矩形单元来跨层细化；一种允许网格间距远离边界层呈指数增长；另一种采用基于Zienkiewicz和Zhu误差估计器的自适应策略。在四个高各向异性二维测试案例中，自适应网格细化策略始终能以降低数个量级的自由度达到与均匀细化相同的精度。在磁场与网格对齐的测试案例中，其他细化策略也显示出显著的效率提升。本研究还讨论了将结果推广至更大磁各向异性比及三维问题的可能性。结果表明：各向同性网格细化所需自由度在二维中与边界层宽度成正比，在三维中与边界层宽度的平方成正比；而各向异性细化在所有维度中所需自由度与边界层宽度的对数成正比。此外，使用代数多重网格预条件时，共轭梯度迭代次数随边界层宽度呈幂次增长，而使用ILU预条件时，迭代次数与边界层宽度无关。