In this paper, we propose a robust multigrid method for 1D immersed finite element method (IFEM). It is shown that the multigrid method is optimal, which means that the convergence rate of the multigrid method is not only independent of the mesh size h and mesh level L, but also independent of the jump of the discontinuous coefficients. Although we only consider 1D interface method, to the best of our knowledge, this is the first attempt to give a rigorous theoretical analysis for the multigrid method for the IFEM. On the way to this goal, we also revisit the IFEM for the 1D interface problem and prove that the error estimates with respect to the L² norm and weighted H¹ semi‐norm are optimal and independent of the jump of the discontinuous coefficients. Numerical results are given to verify our theoretical findings.

中文翻译：

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一维沉浸式有限元方法的鲁棒多重网格方法

在本文中，我们为一维沉浸式有限元方法（IFEM）提出了一种鲁棒的多网格方法。结果表明，多网格方法是最优的，这意味着多网格方法的收敛速度不仅与网格尺寸h和网格级别L无关，而且与不连续系数的跳跃无关。尽管我们仅考虑一维界面方法，但据我们所知，这是对IFEM的多网格方法进行严格理论分析的首次尝试。在朝着这个目标前进的过程中，我们还重新审视了IFEM以解决一维界面问题，并证明了有关L ²范数和加权H ¹的误差估计半范数是最优的，并且与不连续系数的跳跃无关。数值结果证明了我们的理论发现。