Abstract We introduce a new geometric multigrid algorithm to solve elliptic interface problems. First we discretize the problems by the usual P 1 {P_{1}} -conforming finite element methods on a semi-uniform grid which is obtained by refining a uniform grid. To solve the algebraic system, we adopt subspace correction methods for which we use uniform grids as the auxiliary spaces. To enhance the efficiency of the algorithms, we define a new transfer operator between a uniform grid and a semi-uniform grid so that the transferred functions satisfy the flux continuity along the interface. In the auxiliary space, the system is solved by the usual multigrid algorithm with a similarly modified prolongation operator. We show 𝒲 {\mathcal{W}} -cycle convergence for the proposed multigrid algorithm. We demonstrate the performance of our multigrid algorithm for problems having various ratios of parameters. We observe that the computational complexity of our algorithms are robust for all problems we tested.

中文翻译：

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一种求解椭圆界面问题的半均匀多重网格算法

摘要 我们介绍了一种新的几何多重网格算法来解决椭圆界面问题。首先，我们在半均匀网格上通过通常的 P 1 {P_{1}} 符合有限元方法对问题进行离散化，该网格是通过细化均匀网格获得的。为了解决代数系统，我们采用子空间校正方法，我们使用均匀网格作为辅助空间。为了提高算法的效率，我们在均匀网格和半均匀网格之间定义了一个新的传递算子，以便传递的函数满足沿界面的通量连续性。在辅助空间中，系统通过带有类似修改的扩展算子的常用多重网格算法求解。我们展示了提出的多重网格算法的 𝒲 {\mathcal{W}} -循环收敛。我们展示了我们的多重网格算法对于具有各种参数比率的问题的性能。我们观察到我们算法的计算复杂性对于我们测试的所有问题都是稳健的。