A multiscale hybrid-mixed method for the Helmholtz equation in heterogeneous domains

This work proposes a novel multiscale finite element method for acoustic wave propagation in highly heterogeneous media which is accurate on coarse meshes. It originates from the primal hybridization of the Helmholtz equation at the continuous level, which relaxes the continuity of the unknown on the skeleton of a partition. As a result, face-based degrees of freedom drive the approximation on the faces, and independent local problems respond for the multiscale basis function computation. We show how to recover other well-established numerical methods when the basis functions are promptly available. A numerical analysis of the method establishes its well-posedness and proves the quasi-optimality for the numerical solution. Also, we demonstrate that the MHM method is super-convergent in natural norms. We assess theoretical results, as well as the performance of the method on heterogeneous domains, through a sequence of numerical tests