Volume 32, Issue 5
Poisson Preconditioning for Self-Adjoint Elliptic Problems

J. Comp. Math., 32 (2014), pp. 560-578.

Published online: 2014-10

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• Abstract

In this paper, we formulate interface problem and Neumann elliptic boundary value problem into a form of linear operator equations with self-adjoint positive definite operators. We prove that in the discrete level the condition number of these operators is independent of the mesh size. Therefore, given a prescribed error tolerance, the classical conjugate gradient algorithm converges within a fixed number of iterations. The main computation task at each iteration is to solve a Dirichlet Poisson boundary value problem in a rectangular domain, which can be furnished with fast Poisson solver. The overall computational complexity is essentially of linear scaling.

• Keywords

Fast Poisson solver Interface problem Self-adjoint elliptic problem Conjugate gradient method

• AMS Subject Headings

65N30 65T50.

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@Article{JCM-32-560, author = {Houde Han and Chunxiong Zheng}, title = {Poisson Preconditioning for Self-Adjoint Elliptic Problems}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {5}, pages = {560--578}, abstract = {

In this paper, we formulate interface problem and Neumann elliptic boundary value problem into a form of linear operator equations with self-adjoint positive definite operators. We prove that in the discrete level the condition number of these operators is independent of the mesh size. Therefore, given a prescribed error tolerance, the classical conjugate gradient algorithm converges within a fixed number of iterations. The main computation task at each iteration is to solve a Dirichlet Poisson boundary value problem in a rectangular domain, which can be furnished with fast Poisson solver. The overall computational complexity is essentially of linear scaling.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1405-m4293}, url = {http://global-sci.org/intro/article_detail/jcm/9904.html} }
TY - JOUR T1 - Poisson Preconditioning for Self-Adjoint Elliptic Problems AU - Houde Han & Chunxiong Zheng JO - Journal of Computational Mathematics VL - 5 SP - 560 EP - 578 PY - 2014 DA - 2014/10 SN - 32 DO - http://doi.org/10.4208/jcm.1405-m4293 UR - https://global-sci.org/intro/article_detail/jcm/9904.html KW - Fast Poisson solver KW - Interface problem KW - Self-adjoint elliptic problem KW - Conjugate gradient method AB -

In this paper, we formulate interface problem and Neumann elliptic boundary value problem into a form of linear operator equations with self-adjoint positive definite operators. We prove that in the discrete level the condition number of these operators is independent of the mesh size. Therefore, given a prescribed error tolerance, the classical conjugate gradient algorithm converges within a fixed number of iterations. The main computation task at each iteration is to solve a Dirichlet Poisson boundary value problem in a rectangular domain, which can be furnished with fast Poisson solver. The overall computational complexity is essentially of linear scaling.

Houde Han & Chunxiong Zheng. (1970). Poisson Preconditioning for Self-Adjoint Elliptic Problems. Journal of Computational Mathematics. 32 (5). 560-578. doi:10.4208/jcm.1405-m4293
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