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 - <p style="text-align: justify;">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.</p>