An adapted bubble approach which is a modifiation of the residual-free bubbles (RFB) method, is proposed for the Helmhotz 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. The AB method 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. We provide analysis to show how the AB method mitigates the pollution error.
翻译：为 2D 中的 Helmhotz 问题提议了一种经调整的泡沫法,这是对剩余无气泡(RFB) 方法的修改。 为气泡函数近似值采用了一种新的两级限制元素法。与其他等式不同,如平面-扩散方程式,适用于Helmholtz 等式时的RFB 方法,并不取决于另一种稳定法,以获得与子问题解决方案的近似值。经调整的气泡(AB) 是通过对子问题进行简单修改获得的。这种修改使数字解决方案的准确性大大提高。AB 方法能够以 2D 有效解决Helmholtz 等式, 直至 c 是波数和 h 是米什大小。我们提供了分析,以显示 AB 方法如何减轻污染错误。
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