The MHM Method for the Helmholtz Equation

Théophile Chaumont-Frelet (1) , Frédéric Valentin (2)

1 BCAM - Basque Center for Applied Mathematics
2 LNCC / MCT - Laboratorio Nacional de Computação Cientifica [Rio de Janeiro]

Wave propagation in heterogeneous media is accurately approximated on coarse meshes through a novel multiscale finite element method. Such a numerical method originates from the primal hybridization of the Helmholtz equation wherein the continuity of the pressure is released on the skeleton of a partition. As a result, the method is driven by the face degree of freedoms defined on faces of the partition, and independent local problems are responsible for the multiscale basis function computation. A two-level version of the method is also proposed in the case the basis functions are not promptly available. Well-posedness and a best approximation result is established for the one-and two-level MHM methods. Also, the MHM method is proved to be super convergent and is shown to recover other numerical methods. We assess theoretical results through a sequence of numerical tests.

Domaines: Mathématiques [math]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Théophile Chaumont-Frelet, Frédéric Valentin. The MHM Method for the Helmholtz Equation. 2018. ⟨hal-01698914v1⟩