An Edge Multiscale Interior Penalty Discontinuous Galerkin method for heterogeneous Helmholtz problems with large varying wavenumber

Highlights: • It can be equiped with state-of-the-art preconditioners with almost linear complexity. • The associated local solvers are well-posed without further requirement on the wavenumber. • A new multiscale space based on local stable decomposition of the solution space is established. • We derive theoretically the influence of the size of the oversampling element on our method. • It is suitable to many practical applications with large varying wavenumbers. We propose an Edge Multiscale Finite Element Method (EMsFEM) based on an Interior Penalty Discontinuous Galerkin (IPDG) formulation for the heterogeneous Helmholtz problems with large wavenumber. A novel local multiscale space is constructed by solving local problems with a mixed boundary condition composed of a nonhomogeneous Dirichlet boundary condition and an absorbing boundary condition, which can capture the local behavior of the wave propagation and local media information. The key ingredient of our method consists of choosing appropriate Dirichlet data inspired by recent development on edge multiscale basis functions [9], [24], [39], [49]. An IPDG formulation is applied to facilitate generating a sparse linear system and to reduce computational complexity. The convergence rate is derived for wavelet-based and polynomial-based edge multiscale basis functions. Extensive numerical tests in two and three dimensional heterogeneous media are presented to show the supreme performance of our method.