In this paper, we present a novel fast method to solve Poisson's equation in an arbitrary two dimensional region with Neumann boundary condition, which are frequently encountered in solving electrostatic boundary problems. The basic idea is to solve the original Poisson's equation by a two-step procedure. In the first stage, we expand the electric field of interest by a set of tree basis functions and solve it with a fast tree solver in O(N) operations. The field such obtained, however, fails to expand the exact field because the tree basis is not curl-free. Despite of this, we can retrieve the correct electric field by purging the divergence-free field. Next, for the second stage, we find the potential distribution rapidly with a same fast solution of O(N) complexity. As a result, the proposed method dramatically reduces solution time compared with traditional FEM with iterative method. In addition, it is the first time that the loop-tree decomposition technique has been introduced to develop fast Poisson solvers. Numerical examples including electrostatic simulations are presented to demonstrate the efficiency of the proposed method.
Weng Cho Chew,
Li Jun Jiang,
"A Novel Fast Solver for Poisson's Equation with Neumann Boundary Condition," Progress In Electromagnetics Research,
Vol. 136, 195-209, 2013. doi:10.2528/PIER12112010
1. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley, Aug. 1998.
2. Barkas, S. N., An introduction to fast poisson solvers, 2005, http://people.freebsd.org/ snb/school/fastpoisson.pdf.
3. Fogolari, F., A. Brigo, and H. Molinari, "The poissonboltzmann equation for biomolecular electrostatics: A tool for structural biology," Journal of Molecular Recognition, Vol. 15, No. 6, 377-392, 2002. doi:10.1002/jmr.577
4. Adelmann, A., P. Arbenz, and Y. Ineichen, "A fast parallel poisson solver on irregular domains applied to beam dynamics simulations," Journal of Computational Physics, Vol. 229, No. 12, 4554-4566, 2010. doi:10.1016/j.jcp.2010.02.022
5. Lai, M. and W. Wang, "Fast direct solvers for poisson equation on 2D polar and spherical geometries," Numerical Methods for Partial Differential Equations, Vol. 18, No. 1, 56-68, Jan. 2002. doi:10.1002/num.1038
6. Huang, Y.-L., J.-G. Liu, and W.-C. Wang, "An FFT based ast poisson solver on spherical shells," Communications in Computational Physics, Vol. 9, No. 3, SI, 649-667, Mar. 2011.
7. Trottenberg, U., C. W. Oosterlee, and A. Schller, Multigrid, Academic Press, 2001.
9. Briggs, L., V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, Philadelphia, 2000.
10. McAdams, A., E. Sifakis, and J. Teran, "A parallel multigrid Poisson solver for fluids simulation on large grids," ACM SIGGRAPH Symposium on Computer Animation, 2010.
11. McKenney, A., L. Greengard, and A. Mayo, "A fast poisson solver for complex geometries," Journal of Computational Physics, Vol. 118, No. 2, 348-355, 1995. doi:10.1006/jcph.1995.1104
12. Ethridge, F. and L. Greengard, "A new fast-multipole accelerated poisson solver in two dimensions,", Vol. 23, No. 3, 741-760, 2001.
13. Langston, M. H., L. Greengard, and D. Zorin, "A free-space adaptive FMM-based PDE solver in three dimensions," Communications in Applied Mathematics and Computational Science, Vol. 6, No. 1, 79-122, 2011. doi:10.2140/camcos.2011.6.79
14. Greengard, L. and J.-Y. Lee, "A direct adaptive Poisson solver of arbitrary order accuracy," Journal of Computational Physics, Vol. 125, No. 2, 415-424, 1996. doi:10.1006/jcph.1996.0103
15. Wilton, D. R. and A. W. Glisson, "On improving the electric field integral equation at low frequencies," 1981 Spring URSI Radio Science Meeting Digest, 24 Los Angeles, CA, Jun. 1981.
16. Mautz, J. and R. Harrington, "An E-field solution for a conducting surface small or comparable to the wavelength," IEEE Transactions on Antennas and Propagation, Vol. 32, No. 4, 330-339, Apr. 1984. doi:10.1109/TAP.1984.1143316
17. Zhao, J.-S. and W. C. Chew, "Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies ," IEEE Transactions on Antennas and Propagation, Vol. 48, No. 10, 1635-1645, Oct. 2000. doi:10.1109/8.899680
18. Wu, W., A. W. Glisson, and D. Kajfez, "A comparison of two low-frequency formulations for the electric field integral equation," Tenth Ann. Rev. Prog. Appl. Comput. Electromag., Vol. 2, 484-491, 1994.
19. Burton, M. and S. Kashyap, "A study of a recent, moment-method algorithm that is accurate to very low frequencies," Appl. Comput. Electromagn. Soc. J., Vol. 10, No. 3, 58-68, Nov. 1995.
20. Bladel, J. G. V., Electromagnetic Fields, Wiley-IEEE Press, Jun. 2007.
21. Chew, W. C., M. S. Tong, and B. Hu, Integral Equations Methods for Electromagnetic and Elastic Waves, Morgan & Claypool, 2008.
22. Vipiana, F., P. Pirinoli, and G. Vecchi, "A multiresolution method of moments for triangular meshes," IEEE Transactions on Antennas and Propagation, Vol. 53, No. 7, 2247-2258, Jul. 2005. doi:10.1109/TAP.2005.850710
23. Van der Vorst, H. A., "Bi-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems," SIAM J. on Scientific Computing, Vol. 13, 631-644, 1992.
24. Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd Ed., Society for Industrial and Applied Mathematics, 2003.
25. Saad, Y. and M. Schultz, "GMRES: A generalized minimal residue algorithm for solving nonsymmetric linear systems," SIAM J. Sci. Stat. Comput., Vol. 7, 856-869, 1986.