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The main difficulty in solving the discrete source or eigenvalue problems ofthe 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 theirdiscrete systems based on Hodge Laplacian on de Rham complex. The spectra ofthe new schemes are Laplace-like. Then many efficient iterative methods andpreconditioning techniques can be applied to them. After getting the solutionsof the auxiliary schemes, the desired solutions of the original systems can berecovered or recognized through some simple operations. We sum these up as anew framework to compute the discrete source and eigenvalue problems of theoperator $ d^*d $ using iterative method. We also investigate twopreconditioners for the auxiliary schemes, ILU-type method and Multigridmethod. Finally, we present plenty of numerical experiments to verify theefficiency of the auxiliary schemes.

Auxiliary iterative schemes for the discrete operators on de Rham complex