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Volume 16, Issue 1
Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D

Adem Kaya

DOI: 10.4208/nmtma.OA-2022-0083

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 26-57.

Published online: 2023-01

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  • Abstract

An adapted-bubbles approach which is a modification of the residual-free bubbles (RFB) method, is proposed for the Helmholtz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted-bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numerical solution impressively. We provide numerical experiments with the AB method up to $ch = 5$ where $c$ is the wavenumber and $h$ is the mesh size. Numerical tests show that the AB method is better by far than higher order methods available in the literature.

  • Keywords

Helmholtz equation, adapted-bubbles, residual-free bubbles, two-level finite element.

  • AMS Subject Headings

65N30, 65N06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-26, author = {Kaya , Adem}, title = {Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {1}, pages = {26--57}, abstract = {

An adapted-bubbles approach which is a modification of the residual-free bubbles (RFB) method, is proposed for the Helmholtz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted-bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numerical solution impressively. We provide numerical experiments with the AB method up to $ch = 5$ where $c$ is the wavenumber and $h$ is the mesh size. Numerical tests show that the AB method is better by far than higher order methods available in the literature.

}, issn = {2079-7338}, doi = {https://doi.org/ 10.4208/nmtma.OA-2022-0083}, url = {http://global-sci.org/intro/article_detail/nmtma/21342.html} }
TY - JOUR T1 - Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D AU - Kaya , Adem JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 26 EP - 57 PY - 2023 DA - 2023/01 SN - 16 DO - http://doi.org/ 10.4208/nmtma.OA-2022-0083 UR - https://global-sci.org/intro/article_detail/nmtma/21342.html KW - Helmholtz equation, adapted-bubbles, residual-free bubbles, two-level finite element. AB -

An adapted-bubbles approach which is a modification of the residual-free bubbles (RFB) method, is proposed for the Helmholtz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted-bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numerical solution impressively. We provide numerical experiments with the AB method up to $ch = 5$ where $c$ is the wavenumber and $h$ is the mesh size. Numerical tests show that the AB method is better by far than higher order methods available in the literature.

Kaya , Adem. (2023). Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D. Numerical Mathematics: Theory, Methods and Applications. 16 (1). 26-57. doi: 10.4208/nmtma.OA-2022-0083
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