An accurate and efficient multiscale finite element method for the Helmholtz equation with high wave number and strong oscillation(高波数强振荡Helmholtz方程的多尺度有限元精确高效解法)

The Helmholtz equation has important applications in many fields such as acoustics and electromagnetics. However, its high wave number and strong oscillation characteristics lead to systematic difficulties in numerical simulations. Applying a novel multiscale finite element method combined with the embedding technique of multiscale basis functions, the strong oscillation microscopic characteristics of the original problem can be effectively captured, and a specific reduced-order finite-dimensional approximation space can be constructed. Compared with traditional methods, the novel method performs more excellently in capturing the details of wave phenomena. Especially in the case of high wave number, by applying local mesh refinement the high precision can still be maintained, and the computational efficiency can be optimized and the computational time can be reduced. Our research results validate that the multiscale finite element method is particularly suitable for dealing with complicated oscillation problems with multiscale characteristics, and it effectively improves the accuracy, stability and computational efficiency of the numerical solution of the Helmholtz equation, thus possessing significant advantages and application potentials.(Helmholtz方程在声学、电磁学等领域有重要应用，而其高波数强振荡特征致使数值求解面临系统性困难。采用新型的多尺度有限元法，结合多尺度基函数嵌入技术，可有效捕捉原问题的强振荡微观特性，进而构建特定的降阶有限维逼近空间。相较于传统方法，多尺度有限元法在捕捉波动现象的细节上表现卓越，尤其在高波数情形下，通过局部细化网格，依然能保持高精度并优化计算效率、缩减计算时间。研究结果表明，多尺度有限元法特别适合处理具有多尺度特性的复杂振荡问题，有力提升了Helmholtz方程数值解的准确性、稳定性和计算效率，展现出显著的优势与应用潜力。)