The main difficulty in solving the discrete source or eigenvalue problems of the operator $ d^*d $ with iterative methods is to deal with its huge kernel, for example, the $ \nabla \times \nabla \times $ and $- \nabla ( \nabla \cdot ) $ operator. In this paper, we construct a kind of auxiliary schemes for their discrete systems based on Hodge Laplacian on de Rham complex. The spectra of the new schemes are Laplace-like. Then many efficient iterative methods and preconditioning techniques can be applied to them. After getting the solutions of the auxiliary schemes, the desired solutions of the original systems can be recovered or recognized through some simple operations. We sum these up as a new framework to compute the discrete source and eigenvalue problems of the operator $ d^*d $ using iterative method. We also investigate two preconditioners for the auxiliary schemes, ILU-type method and Multigrid method. Finally, we present plenty of numerical experiments to verify the efficiency of the auxiliary schemes.

翻译：解决操作员的离散源或元值问题的主要困难在于用迭代方法处理其巨大的内核,例如, $\nabla\ times\ times\ times $和$-nabla (\nabla\cdot) $ 操作员的离散源或元值问题。 在本文中, 我们根据Hodge Laplacecian的关于de Rham 复合物的离散系统, 为操作员的离散系统建造了一种辅助方案。 新方案的光谱类似 Laplace 。 然后, 许多高效的迭接方法和先决条件技术可以应用到它们。 在获得辅助方案解决方案后, 原始系统的预期解决方案可以通过一些简单的操作得到恢复或确认。 我们把这些归结为使用迭代法计算离散源和操作员的元值问题的新框架 $ d ⁇ d。 我们还调查了辅助方案的两个先决条件, ILU 类方法和MUGrid 方法。 最后, 我们提出了大量的数字实验,以核实辅助方案的效率。