Fast Energy-Norm Convergence and High Efficiency in a Multiscale Finite Element Computation for the Helmholtz Equation

This paper develops a fast-convergent and highly efficient multiscale finite element method for the Helmholtz equation. Through the construction of tailored multiscale basis functions, the proposed method is capable of capturing both local fine-scale characteristics and global macroscale behaviors of multiscale solutions. We further systematically elaborate on the associated data structure, in which a global mapping matrix intrinsically encodes rich microscopic information inside each macroscopic element. Rigorous error analysis is then carried out, combining approximation and interpolation error estimates to establish a second-order convergence result in the energy norm, which furnishes a solid theoretical foundation for the accuracy of the proposed scheme. Representative numerical experiments are presented to verify the performance of the method with regard to computational efficiency and solution stability.