Finite elements of class ${𝒞}^1$ are suitable for the computation of magnetohydrodynamics instabilities in tokamak plasmas. In addition, isoparametric approximations allow for a precise alignment of the mesh with the magnetic field line. Mesh alignment is crucial to achieve axisymmetric equilibria accurately. It is also helpful to deal with the anisotropy nature of magnetized plasma flows. In this numerical framework, several practical simulations are now available. They help to understand better the operation of existing devices and predict the optimal strategies for using the international ITER tokamak under construction. However, a mesh-aligned isoparametric representation suffers from the presence of critical points of the magnetic field (magnetic axis, X-point). We here explore a strategy that combines aligned mesh out of the critical points with non-aligned unstructured mesh in a region containing these points. By this strategy, we can avoid highly stretched elements and the numerical difficulties that come with them. The mesh-aligned interpolation uses bi-cubic Hemite-Bézier polynomials on a structured mesh of curved quadrangular elements. On the other hand, we assume reduced cubic Hsieh-Clough-Tocher finite elements on an unstructured triangular mesh. Both meshes overlap, and the resulting formulation is a coupled discrete problem solved iteratively by a suitable one-level Schwarz algorithm. In this paper, we will focus on the Poisson problem on a two-dimensional bounded regular domain. This elliptic equation is a simplified version of the axisymmetric tokamak equilibrium one at the asymptotic limit of infinite major radius (large aspect ratio).

中文翻译：

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二阶椭圆问题复合曲面网格上 C1 类有限元的耦合

类的有限元${𝒞}^1$适用于计算托卡马克等离子体中的磁流体动力学不稳定性。此外，等参近似允许网格与磁场线精确对齐。网格对齐对于准确实现轴对称平衡至关重要。它也有助于处理磁化等离子体流的各向异性性质。在这个数值框架中，现在可以进行几种实用的模拟。它们有助于更好地了解现有设备的运行情况，并预测使用正在建设的国际 ITER 托卡马克装置的最佳策略。然而，网格对齐的等参表示会受到磁场临界点（磁轴、X 点）的影响。我们在这里探索一种策略，将关键点外的对齐网格与包含这些点的区域中的非对齐非结构化网格相结合。通过这种策略，我们可以避免高度拉伸的元素以及随之而来的数值困难。网格对齐插值在弯曲四边形元素的结构化网格上使用双三次 Hemite-Bézier 多项式。另一方面，我们假设非结构化三角网格上的简化三次 Hsieh-Clough-Tocher 有限元。两个网格重叠，所得公式是通过合适的单级 Schwarz 算法迭代解决的耦合离散问题。在本文中，我们将重点关注二维有界正则域上的泊松问题。该椭圆方程是无限长半径（大长径比）渐近极限处的轴对称托卡马克平衡方程的简化版本。