A superconvergent multiscale scheme for the Helmholtz equation in rapidly oscillating heterogeneous media

We present a superconvergent and highly efficient multiscale finite element scheme for solving the Helmholtz equation in heterogeneous media. The proposed scheme is specifically designed to address the challenges posed by the multiscale nature of the Helmholtz equation, which often leads to high computational costs and numerical instabilities in traditional methods. By constructing tailored multiscale basis functions, the proposed approach effectively captures both the local fine-scale features and global macroscale behaviors of the multiscale solution. The fundamental mechanism underlying the multiscale scheme is systematically expounded. Through rigorous error analyses, we derive a second-order superconvergence result in the $H^1$ norm. Additionally, numerical experiments are conducted to validate the performance of the scheme. Both theoretical findings and numerical results confirm that the proposed multiscale scheme not only achieves higher solution accuracy but also exhibits superconvergence rates and significantly reduces computational costs.