We introduce a new hybridizable discontinuous Galerkin (HDG) method for the numerical solution of the Helmholtz equation over a wide range of wave frequencies. Our approach combines the HDG methodology with geometrical optics in a fashion that allows us to take advantage of the strengths of these two methodologies. The phase-based HDG method is devised as follows. First, we enrich the local approximation spaces with precomputed phases which are solutions of the eikonal equation in geometrical optics. Second, we propose a novel scheme that combines the HDG method with ray tracing to compute multivalued solution of the eikonal equation. Third, we utilize the proper orthogonal decomposition to remove redundant modes and obtain locally orthogonal basis functions which are then used to construct the global approximation spaces of the phase-based HDG method. And fourth, we propose an appropriate choice of the stabilization parameter to guarantee stability and accuracy for the proposed method. Numerical experiments presented show that optimal orders of convergence are achieved, that the number of degrees of freedom to achieve a given accuracy is independent of the wave number, and that the number of unknowns required to achieve a given accuracy with the proposed method is orders of magnitude smaller than that with the standard finite element method.
Bibliographical noteFunding Information:
N.C. Nguyen and J. Peraire would like to acknowledge the partial support by AFOSR Grant No. FA9550-11-1-0141 , AFOSR Grant No. FA9550-12-0357 , and the Singapore-MIT Alliance for Research and Technology Centre . B. Cockburn was partially supported by the National Science Foundation (Grant DMS-0712955 ) and by the Minnesota Supercomputing Institute .
- Discontinuous Galerkin methods
- Eikonal equation
- Finite element method
- Geometrical optics
- Helmholtz equation
- Hybrid/mixed methods