A stable discontinuous Galerkin-type method for solving efficiently Helmholtz problems

We propose a stable discontinuous Galerkin-type method (SDGM) for solving efficiently Helmholtz problems. This mixed-hybrid formulation is a two-step procedure. Step 1 consists in solving well-posed problems at the element partition level of the computational domain, whereas Step 2 requires the solution of a global system whose unknowns are the Lagrange multipliers. The main features of SDGM include: (a) the resulting local problems are associated with small positive definite Hermitian matrices that can be solved in parallel, and (b) the matrix corresponding to the global linear system arising in Step 2 is Hermitian and positive semi-definite. Illustrative numerical results for two-dimensional waveguide and scattering problems highlight the potential of SDGM for solving efficiently Helmholtz problems in mid- and high-frequency regime.