A hybridizable discontinuous Galerkin method with characteristic variables for Helmholtz problems

TL;DR

This paper introduces the CHDG method, a hybridizable discontinuous Galerkin approach utilizing characteristic variables for efficient and improved solution of Helmholtz wave propagation problems, reducing system complexity and enhancing iterative solver performance.

Contribution

The paper presents a novel CHDG method that reduces system size and improves properties over standard HDG for Helmholtz problems, facilitating more efficient iterative solutions.

Findings

Reduced condition number with CHDG

Fewer iterations needed with CGNR or GMRES

Improved properties of the reduced system

Abstract

A new hybridizable discontinuous Galerkin method, named the CHDG method, is proposed for solving time-harmonic scalar wave propagation problems. This method relies on a standard discontinuous Galerkin scheme with upwind numerical fluxes and high-order polynomial bases. Auxiliary unknowns corresponding to characteristic variables are defined at the interface between the elements, and the physical fields are eliminated to obtain a reduced system. The reduced system can be written as a fixed-point problem that can be solved with stationary iterative schemes. Numerical results with 2D benchmarks are presented to study the performance of the approach. Compared to the standard HDG approach, the properties of the reduced system are improved with CHDG, which is more suited for iterative solution procedures. The condition number of the reduced system is smaller with CHDG than with the standard HDG method. Iterative solution procedures with CGNR or GMRES required smaller numbers of iterations with CHDG.