A Sign-Definite Preconditioned High-Order FEM, Part I: Formulation and Simulation for Bounded Homogeneous Media Wave Propagation

In this article, we develop a novel computer model for the constant coefficient Helmholtz equation based on a sign-definite formulation [A. Moiola and E. A. Spence, SIAM Rev., 56 (2014), pp. 274--312], finite element methods (FEMs), and iterative solvers. This includes (i) a variant parameter version of the formulation for improved performance of the GMRES iterates compared to the standard formulation; (ii) development of a new frequency-robust sign-definite preconditioned high-order FEM model for the simulation of acoustic wave propagation in bounded homogeneous media; and (iii) efficient use of domain decomposition methods in conjunction with our preconditioned model. For simulations, we use a nonconvex star-shaped geometry comprising curved and nonsmooth boundaries and a non-star-shaped nonsmooth geometry and demonstrate our approach for medium- to high-frequency incident waves.