A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime Original Research Article

We present a discontinuous Galerkin method (DGM) for the solution of the Helmholtz equation in the mid-frequency regime. Our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non-conforming field that contains free-space solutions of the homogeneous partial differential equation to be solved. Hence, for the Helmholtz equation, the enrichment field is chosen here as the superposition of plane waves. We enforce a weak continuity of these plane waves across the element interfaces by suitable Lagrange multipliers. Preliminary results obtained for two-dimensional model problems discretized by uniform meshes reveal that the proposed DGM enables the development of elements that are far more competitive than both the standard linear and the standard quadratic Galerkin elements for the discretization of Helmholtz problems.