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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

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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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