Abstract
This paper considers finite element discretizations of the Helmholtz equation and its generalization arising from harmonic acoustic perturbations to a nonuniform steady potential flow. A novel elliptic, positive-definite preconditioner with a multigrid implementation is used to accelerate the iterative convergence of Krylov subspace solvers. Both theory and numerical results show that for a model 1-D Helmholtz test problem, the preconditioner clusters the discrete system's eigenvalues and lowers its condition number to a level independent of grid resolution. For the 2-D Helmholtz equation, grid-independent convergence is achieved using a quasi-minimal residual Krylov solver, significantly outperforming the popular symmetric successive over-relaxation preconditioner. Impressive results are also presented on more complex domains, including an axisymmetric aircraft engine inlet with nonstagnant mean flow and modal boundary conditions.
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