We present a multigrid method for an unfitted finite element discretization of the Dirichlet boundary value problem. The discretization employs Nitsche's method to implement the boundary condition and additional face based ghost penalties for stabilization. We apply standard intergrid operators, relying on the fact that the relevant domain of computation does not grow under mesh refinement. The smoother is a parallel implementation of the multiplicative vertex-patch smoother with inconsistent treatment of ghost penalties. Our computational results show that we obtain a fast converging method. Furthermore, runtime comparison to fitted methods show that the losses are moderate although many optimizations for Cartesian vertex patches cannot be applied on cut patches.

翻译：本文针对狄利克雷边值问题的非拟合有限元离散化提出了一种多重网格方法。该离散化方案采用Nitsche方法实现边界条件，并引入基于单元面的虚影罚项以增强稳定性。我们采用标准网格间传递算子，其可行性基于计算相关域在网格细化过程中不会扩展的特性。平滑器采用乘法型顶点块平滑器的并行实现，其中对虚影罚项进行非一致性处理。计算结果表明，该方法具有快速收敛特性。此外，与拟合方法的运行时间对比显示，尽管笛卡尔顶点块的多种优化技术无法直接应用于切割块，但计算效率损失仍在可接受范围内。