New Multilevel Preconditioner for Diffusion-type Problems on Polyhedral Meshes

Daniil Svyatskiy

Email: dasvyat@lanl.gov
Postal Address: Mathematical Modeling and Analysis, Theoretical Division, Mail Stop B284, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.

We present a new approach to the construction of two-level spectrally equivalent preconditioner for diffusion type problems on polygonal or polyhedral meshes. Assuming some coarsening procedure this approach can be directly extended to multilevel framework and can be parallelized without any significant modifications.

The main idea of our approach is to define the grid transfer operators in such a  way that the realization is cheap from computational point of view and the resulting preconditioner is spectrally equivalent to the matrix of the system. We admit that the iteration number of PCG method with our preconditioner can be greater with compare to other preconditioners but since the cost of the initialization procedure and the iteration cost are relatively cheap we observe significant savings in the computational time.

The numerical experiments show that this preconditioner is competitive with compare to such popular approach as algebraic multigrid (AMG).