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Volume 5, Issue 6
A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions

Long Yuan & Qiya Hu

Adv. Appl. Math. Mech., 5 (2013), pp. 791-808.

Published online: 2013-05

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  • Abstract

An interesting discretization method for Helmholtz equations was introduced in B. Després [1]. This method is based on the ultra weak variational formulation (UWVF) and the wave shape functions, which are exact solutions of the governing Helmholtz equation. In this paper we are concerned with fast solver for the system generated by the method in [1]. We propose a new preconditioner for such system, which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in [13]. In our numerical experiments, this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations, and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.

  • Keywords

Helmholtz equation, ultra weak variational formulation, wave shape functions, preconditioner, iteration counts.

  • AMS Subject Headings

65F08, 65N55

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-5-791, author = {Yuan , Long and Hu , Qiya}, title = {A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {6}, pages = {791--808}, abstract = {

An interesting discretization method for Helmholtz equations was introduced in B. Després [1]. This method is based on the ultra weak variational formulation (UWVF) and the wave shape functions, which are exact solutions of the governing Helmholtz equation. In this paper we are concerned with fast solver for the system generated by the method in [1]. We propose a new preconditioner for such system, which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in [13]. In our numerical experiments, this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations, and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.12-m12142}, url = {http://global-sci.org/intro/article_detail/aamm/96.html} }
TY - JOUR T1 - A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions AU - Yuan , Long AU - Hu , Qiya JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 791 EP - 808 PY - 2013 DA - 2013/05 SN - 5 DO - http://doi.org/10.4208/aamm.12-m12142 UR - https://global-sci.org/intro/article_detail/aamm/96.html KW - Helmholtz equation, ultra weak variational formulation, wave shape functions, preconditioner, iteration counts. AB -

An interesting discretization method for Helmholtz equations was introduced in B. Després [1]. This method is based on the ultra weak variational formulation (UWVF) and the wave shape functions, which are exact solutions of the governing Helmholtz equation. In this paper we are concerned with fast solver for the system generated by the method in [1]. We propose a new preconditioner for such system, which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in [13]. In our numerical experiments, this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations, and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.

Long Yuan & Qiya Hu. (1970). A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions. Advances in Applied Mathematics and Mechanics. 5 (6). 791-808. doi:10.4208/aamm.12-m12142
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