Sign-definite formulations and frequency-robust preconditioned FEMs: Part 1

Abstract: 

The standard Galerkin formulation of the acoustic wave propagation governed by the Helmholtz partial differential equation (PDE) is indefinite for large wavenumbers. However, the Helmholtz PDE  is in general not indefinite.  The lack of coercivity (indefiniteness) in the formulation  and associated finite element method (FEM) models is one of the major difficulties for approximation and simulation of wave propagation models using iterative methods.  

We will present (in two talks) a new class of constructive preconditioned sign-definite FEM Helmholtz wave propagation models in homogeneous and heterogeneous media. The main focus of Part I will be on the constant coefficient Helmholtz PDE; and Part II will deal with heterogeneous media, coercive formulations and analysis.  

Our new preconditioned FEM formulations provide concrete answers to some key issues raised by the authors in a recent SIAM Review article about the practical use of their theoretical homogeneous media sign-definite formulation. Further, we remove the notion of sign-indefiniteness and frequency-sensitive iterations associated with heterogeneous media acoustic wave propagation models.