A new two-level finite element method is introduced for the approximations of the residual-free bubble (RFB) functions and its application to the Helmholtz equation with large wave numbers is considered. Although this approach was considered for the Helmholtz equation before, our new insights show that some of its important properties have remained hidden. 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. Furthermore, it is possible to further increase the accuracy of the solutions in 2D by increasing the support of the integrals containing the bubble functions. The modified-RFB is able to solve the Helmholtz equation efficiently in 2D up to ch = 3.5 where c is the wave number and h is the mesh size.

中文翻译：

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无残差气泡在大波数的亥姆霍兹方程中的应用

引入了一种新的两级有限元方法来近似无残差气泡（RFB）函数，并考虑将其应用于具有大波数的Helmholtz方程。尽管以前在Helmholtz方程中考虑过这种方法，但我们的新见识表明，它的一些重要属性仍然被隐藏。与平流扩散方程等其他方程式不同，RFB方法应用于亥姆霍兹方程式时，不依赖于另一种稳定化方法来获得子问题解的近似值。此外，可以通过增加包含气泡函数的积分的支持来进一步提高2D解的精度。修改后的RFB能够有效地在ch = 3的情况下以二维方式求解Helmholtz方程。