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A Fast Direct Solver for a Class of 3-D Elliptic Partial Differential Equation with Variable Coefficient

Published online by Cambridge University Press:  20 August 2015

Beibei Huang ,
Bin Tu  and
Benzhuo Lu
Beibei Huang*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Academy of Science, Beijing 100190, P.R. China Departament d’Enginyeria Química, Universitat Rovira i Virgili, Av. dels Päısos Catalans, 26 Tarragona 43007, Spain
Bin Tu*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Academy of Science, Beijing 100190, P.R. China
Benzhuo Lu*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Academy of Science, Beijing 100190, P.R. China
*
Corresponding author.Email:hbb21st@lsec.cc.ac.cn
Email:tubin@lsec.cc.ac.en
Email:bzlu@lsec.cc.ac.en
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Abstract

We propose a direct solver for the three-dimensional Poisson equation with a variable coefficient, and an algorithm to directly solve the associated sparse linear systems that exploits the sparsity pattern of the coefficient matrix. Introducing some appropriate finite difference operators, we derive a second-order scheme for the solver, and then two suitable high-order compact schemes are also discussed. For a cube containing N nodes, the solver requires arithmetic operations and memory to store the necessary information. Its efficiency is illustrated with examples, and the numerical results are analysed.

Keywords

15A1515A0915A23Fast solverdirect methoddiscrete Laplace operatorfast matrix inversion
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 4 , October 2012 , pp. 1148 - 1162
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Martinsson, P.‐G., A fast direct solver for a class of elliptic partial differential equations, J. Sci. Comput. 38 (2009), pp. 316–330.Google Scholar
[2]Engquist, B. and Ying, L., Sweeping preconditioner for the Helmholtz equation: Moving perfectly matched layers, Multiscale Model. Sim. 9(2) (2011), 686–710.Google Scholar
[3]Chandrasekaran, S., Gu, M., and Lyons, W., A fast adaptive solver for hierarchically semisep-arable representations, Calcolo 42(3-4) (2005), pp. 171–185.Google Scholar
[4]Chandrasekaran, S., Gu, M., and Lyons, W., Superfast multifrontal method for structured linear systems of equations, Private Communication (2007).Google Scholar
[5]Grasedyck, L., Kriemann, R., and Le Borne, S., Domain-decomposition based H-matrix pre-conditioners, Proceedings of DD16. LNSCE, Vol. 55, Springer, Berlin (2006), pp. 661–668, pp. 471–476.Google Scholar
[6]Le Tallec, P., Domain decomposition methods in computational mechanical, Computational Mechanics Advances (1986), pp. 121–220.Google Scholar
[7]Pack, G.R., Garrett, G.A., Wong, L., and Lamm, G., The Effect of a variable dielectric coefficient and finite Ion size on Poisson-Boltzmann calculations of DNA-electrolyte systems, Biophysical J. 65 (1993), pp. 1363–1370.Google Scholar
[8]Warwicker, J. and Watson, H.C., Calculation of the electric potential in the active site cleft due to α-helical dipoles, J. Mol. Biol. 157 (1982), pp. 671–679.CrossRefGoogle Scholar
[9]Lu, B.Z. and McCammon, J.A., Molecular surface-free continuum model for electrodiffusion processes, Chem. Phys. Lett. 451(4-6) 2008, pp. 282–286.Google Scholar
[10]Warming, R.F. and Hyett, B.J., The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys. 14(2) (1974), pp. 159–179.Google Scholar
[11]Hackbusch, W., Elliptic Differential Equations Theory and Numerical Treatment, Springer-Verlag Berlin Heidelberg, Berlin, 1992, pp. 40–72. (Electronic).Google Scholar
[12]Iserles, A., A First Course in the Numerical Analysis of Differential Equations, Tinghua University Press, 2005, pp. 101–127. (Chinese version).Google Scholar
[13]Yu, D.H. and Tang, H.Z., Numerical Solution of Differential Equations, Science Press, 2007, pp. 300–307. (Chinese version).Google Scholar
[14]Liberty, E., Woolfe, F., Martinsson, P., Rokhlin, V., and Tygert, M., Randomized algorithms for the low-rank approximation of matrices, Proc. Natl. Acad. Sci. USA 104 (2007), pp. 20167– 0172.Google Scholar
[15]Woolfe, F., Liberty, E., Rokhlin, V., and Tygert, M., A fast randomized algorithm for the approximation of matrices, Appl. Comput. Harmon. Anal. 25 (2008), pp. 335–366.Google Scholar
[16]Lin, L., Lu, J., and Ying, L., Fast construction of hierarchical matrix representation from matrix-vector multiplication, J. Comput. Phys. 230(10) (2010), pp. 4071–4087.Google Scholar
[17]Spotz, W.F. and Carey, G.F., A high-order compact formulation for the 3-D Poisson equation, Numer. Methods Part. Diff. Eqs. 12 (1996), pp. 235–243.Google Scholar
[18]Spotz, W. F. and Carey, G. F., High-order compact scheme for the steady stream-function vorticity equations, Int. J. Numer. Methods Eng. 38 (1995), pp. 3497–3512.CrossRefGoogle Scholar
[19]Gupta, M.M. and Kouatchou, J., Symbolic derivation of finite difference approximations for three dimensional Poisson equation, Int. J. Numer. Methods Part. Diff. Eqs. 14(5) (1998), pp. 593–606.Google Scholar
[20]Ananthakrishnaiah, U., Manohar, R., and Stephenson, J. W., Fourth-order finite difference for three-dimensional general elliptic problems with variable coefficients, Numer. Methods Part. Diff. Eqs. 3 (1987), pp. 229–240.Google Scholar
[21]Zhang, J., An explicit fourth-order compact finite difference scheme for three dimensional convection-diffusion equation, Commun. Numer. Methods Eng. 14 (1997), pp. 263–280.Google Scholar