Commun. Comput. Phys., 12 (2012), pp. 1148-1162. |
A Fast Direct Solver for a Class of 3-D Elliptic Partial Differential Equation with Variable Coefficient Beibei Huang ^{1*}, Bin Tu ^{2}, Benzhuo Lu ^{2} 1 Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Academy of Science, Beijing 100190, P.R. China; and Departament d'Enginyeria Quimica, Universitat Rovira i Virgili, Av. dels Paisos Catalans, 26 Tarragona 43007, Spain.2 Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Academy of Science, Beijing 100190, P.R. China. Received 10 November 2010; Accepted (in revised version) 6 December 2011 Available online 17 April 2012 doi:10.4208/cicp.101110.061211a 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 O(N^{3/2}log^{2}N) arithmetic operations and O(N log N) memory to store the necessary information. Its efficiency is illustrated with examples, and the numerical results are analysed. AMS subject classifications: 15A15, 15A09, 15A23Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Fast solver, direct method, discrete Laplace operator, fast matrix inversion. *Corresponding author. Email: hbb21st@lsec.cc.ac.cn (B. Huang), tubin@lsec.cc.ac.cn (B. Tu), bzlu@lsec.cc.ac.cn (B. Lu) |