Volume 35, Issue 1
A Saddle Point Numerical Method for Helmholtz Equations

Russell B. Richins

J. Comp. Math., 35 (2017), pp. 19-36.

Published online: 2017-02

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  • Abstract

In a previous work, the author and D.C. Dobson proposed a numerical method for solving the complex Helmholtz equation based on the minimization variational principles developed by Milton, Seppecher, and Bouchitté. This method results in a system of equations with a symmetric positive definite coefficient matrix, but at the same time requires solving simultaneously for the solution and its gradient. Herein is presented a method based on the saddle point variational principles of Milton, Seppecher, and Bouchitté, which produces symmetric positive definite systems of equations, but eliminates the necessity of solving for the gradient of the solution. The result is a method for a wide class of Helmholtz problems based completely on the Conjugate Gradient algorithm.

  • Keywords

Helmholtz Conjugate gradient Saddle point Finite element

  • AMS Subject Headings

Primary 65N30 Secondary 35A15.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

rrichins@thiel.edu (Russell B. Richins)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-19, author = {Richins , Russell B. }, title = {A Saddle Point Numerical Method for Helmholtz Equations}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {1}, pages = {19--36}, abstract = { In a previous work, the author and D.C. Dobson proposed a numerical method for solving the complex Helmholtz equation based on the minimization variational principles developed by Milton, Seppecher, and Bouchitté. This method results in a system of equations with a symmetric positive definite coefficient matrix, but at the same time requires solving simultaneously for the solution and its gradient. Herein is presented a method based on the saddle point variational principles of Milton, Seppecher, and Bouchitté, which produces symmetric positive definite systems of equations, but eliminates the necessity of solving for the gradient of the solution. The result is a method for a wide class of Helmholtz problems based completely on the Conjugate Gradient algorithm.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1604-m2014-0136}, url = {http://global-sci.org/intro/article_detail/jcm/9761.html} }
TY - JOUR T1 - A Saddle Point Numerical Method for Helmholtz Equations AU - Richins , Russell B. JO - Journal of Computational Mathematics VL - 1 SP - 19 EP - 36 PY - 2017 DA - 2017/02 SN - 35 DO - http://doi.org/10.4208/jcm.1604-m2014-0136 UR - https://global-sci.org/intro/article_detail/jcm/9761.html KW - Helmholtz KW - Conjugate gradient KW - Saddle point KW - Finite element AB - In a previous work, the author and D.C. Dobson proposed a numerical method for solving the complex Helmholtz equation based on the minimization variational principles developed by Milton, Seppecher, and Bouchitté. This method results in a system of equations with a symmetric positive definite coefficient matrix, but at the same time requires solving simultaneously for the solution and its gradient. Herein is presented a method based on the saddle point variational principles of Milton, Seppecher, and Bouchitté, which produces symmetric positive definite systems of equations, but eliminates the necessity of solving for the gradient of the solution. The result is a method for a wide class of Helmholtz problems based completely on the Conjugate Gradient algorithm.
Russell B. Richins . (2020). A Saddle Point Numerical Method for Helmholtz Equations. Journal of Computational Mathematics. 35 (1). 19-36. doi:10.4208/jcm.1604-m2014-0136
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