Abstract The linear complementarity problem arising from a free boundary problem can be equivalently reformulated as a fixed-point equation. We present a modified modulus-based multigrid method to solve this fixed-point equation. This modified method is a full approximation scheme using the modulus-based splitting iteration method as the smoother and avoids the transformation between the auxiliary and the original functions which was necessary in the existing modulus-based multigrid method. We predict its asymptotic convergence factor by applying local Fourier analysis to the corresponding two-grid case. Numerical results show that the W-cycle possesses an h-independent convergence rate and a linear elapsed CPU time, and the convergence rate of the V-cycle can be improved by increasing the smoothing steps. Compared with the existing modulus-based multigrid method, the modified method is more straightforward and is a standard full approximation scheme, which makes it more convenient and efficient in practical applications.

中文翻译：

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自由边界问题引起的线性互补问题的一种修正的基于模数的多重网格方法

摘要 由自由边界问题引起的线性互补问题可以等价地重新表述为不动点方程。我们提出了一种改进的基于模数的多重网格方法来求解这个定点方程。这种改进的方法是一种完全近似方案，使用基于模数的分裂迭代方法作为平滑器，避免了现有基于模数的多重网格方法中必须进行的辅助函数和原始函数之间的转换。我们通过将局部傅立叶分析应用于相应的双网格情况来预测其渐近收敛因子。数值结果表明，W-cycle具有与h无关的收敛速度和线性的CPU时间，通过增加平滑步长可以提高V-cycle的收敛速度。