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Volume 16, Issue 3
Analysis of Pollution-Free Approaches for Multi-Dimensional Helmholtz Equations

Kun Wang, Yau Shu Wong & Jizu Huang

Int. J. Numer. Anal. Mod., 16 (2019), pp. 412-435.

Published online: 2018-11

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  • Abstract

Motivated by our recent work about pollution-free difference schemes for solving Helmholtz equation with high wave numbers, this paper presents an analysis of error estimate for the numerical solution on the annulus and hollow sphere domains. By applying the weighted-test-function method and defining two special interpolation operators, we first derive the existence, uniqueness, stability and the pollution-free error estimate for the one-dimensional problems generated from a method based on separation of variables. Utilizing the spherical harmonics and approximations results, we then prove the pollution-free error estimate in $L^2$-norm for multi-dimensional Helmholtz problems.

  • Keywords

Helmholtz equation, error estimate, finite difference method, polar and spherical coordinates, pollution-free scheme.

  • AMS Subject Headings

65N06, 65N15, 65N22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

kunwang@cqu.edu.cn (Kun Wang)

yaushu.wong@ualberta.ca (Yau Shu Wong)

huangjz@lsec.cc.ac.cn (Jizu Huang)

  • BibTex
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  • TXT
@Article{IJNAM-16-412, author = {Wang , KunWong , Yau Shu and Huang , Jizu}, title = {Analysis of Pollution-Free Approaches for Multi-Dimensional Helmholtz Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {16}, number = {3}, pages = {412--435}, abstract = {

Motivated by our recent work about pollution-free difference schemes for solving Helmholtz equation with high wave numbers, this paper presents an analysis of error estimate for the numerical solution on the annulus and hollow sphere domains. By applying the weighted-test-function method and defining two special interpolation operators, we first derive the existence, uniqueness, stability and the pollution-free error estimate for the one-dimensional problems generated from a method based on separation of variables. Utilizing the spherical harmonics and approximations results, we then prove the pollution-free error estimate in $L^2$-norm for multi-dimensional Helmholtz problems.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12876.html} }
TY - JOUR T1 - Analysis of Pollution-Free Approaches for Multi-Dimensional Helmholtz Equations AU - Wang , Kun AU - Wong , Yau Shu AU - Huang , Jizu JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 412 EP - 435 PY - 2018 DA - 2018/11 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12876.html KW - Helmholtz equation, error estimate, finite difference method, polar and spherical coordinates, pollution-free scheme. AB -

Motivated by our recent work about pollution-free difference schemes for solving Helmholtz equation with high wave numbers, this paper presents an analysis of error estimate for the numerical solution on the annulus and hollow sphere domains. By applying the weighted-test-function method and defining two special interpolation operators, we first derive the existence, uniqueness, stability and the pollution-free error estimate for the one-dimensional problems generated from a method based on separation of variables. Utilizing the spherical harmonics and approximations results, we then prove the pollution-free error estimate in $L^2$-norm for multi-dimensional Helmholtz problems.

Wang , KunWong , Yau Shu and Huang , Jizu. (2018). Analysis of Pollution-Free Approaches for Multi-Dimensional Helmholtz Equations. International Journal of Numerical Analysis and Modeling. 16 (3). 412-435. doi:
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