MILU preconditioning is known to be the optimal one among all the ILU-type preconditionings in solving the Poisson equation with Dirichlet boundary condition. It is optimal in the sense that it reduces the condition number from O(h−2), which can be obtained from other ILU-type preconditioners, to O(h−1). However, with Neumann boundary condition, the conventional MILU cannot be used since it is not invertible, and some MILU preconditionings achieved the order O(h−1) only in rectangular domains. In this article, we consider a standard finite volume method for solving the Poisson equation with Neumann boundary condition in general smooth domains, and introduce a new and efficient MILU preconditioning for the method in two dimensional general smooth domains. Our new MILU preconditioning achieved the order O(h−1) in all our empirical tests. In addition, in a circular domain with a fine grid, the CG method preconditioned with the proposed MILU runs about two times faster than the CG with ILU.
Bibliographical noteFunding Information:
The research of C. Min and Y. Park was supported by Basic Science Research Program through the National Re-search Foundation of Korea (NRF) funded by the Ministry of Education ( 2017-006688 ) and the research of J. Kim, J. Jung and E. Lee was supported by the Samsung Science and Technology Foundation under Project (Number SSTF-BA1301-03 ).
© 2017 Elsevier Inc.
- Finite volume method
- MILU preconditioning
- Neumann boundary condition
- Poisson equation
- Purvis–Burkhalter method