• Corpus ID: 243832729

A note on using the mass matrix as a preconditioner for the Poisson equation

@article{Greif2021ANO,
  title={A note on using the mass matrix as a preconditioner for the Poisson equation},
  author={Chen Greif and Yunhui He},
  journal={ArXiv},
  year={2021},
  volume={abs/2111.03191}
}
We show that the mass matrix derived from finite elements can be effectively used as a preconditioner for iteratively solving the linear system arising from finite-difference discretization of the Poisson equation, using the conjugate gradient method. We derive analytically the condition number of the preconditioned operator. Theoretical analysis shows that the ratio of the condition number of the Laplacian to the preconditioned operator is 8{3 in one dimension, 9{2 in two dimensions, and 2{3… 

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References

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A closed-form multigrid smoothing factor for an additive Vanka-type smoother applied to the Poisson equation

The mass matrix obtained from finite element method can be used as an approximation to the inverse of the Laplacian, and the resulting mass-based relaxation scheme features small smoothing factors in one, two, and three dimensions.

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