Preconditioning a Finite Element Solver of the Exterior Helmholtz Equation

We consider acoustic scattering about a general body. This is described by the Helmholtz equation exterior to the body. In order to truncate the infinite domain we use the BGT absorbing boundary condition. The resultant problem in a finite domain is solved by a finite element procedure. This yields a large sparse system of linear equations which is neither symmetric nor positive definite. We solve the system by an iterative Krylov space type method. To increase the rate of convergence a preconditioner is introduced. This preconditioner is based on a different Helmholtz equation with complex coefficients. This preconditioned system is again solved by a Krylov space method with an ILU preconditioner. Computations are presented to show the efficiency of this technique

Convergence Acceleration for the Three Dimensional Compressible Navier-Stokes Equations

We consider a multistage algorithm to advance in pseudo-time to find a steady state solution for the compressible Navier-Stokes equations. The rate of convergence to the steady state is improved by using an implicit preconditioner to approximate the numerical scheme. This properly addresses the stiffness in the discrete equations associated with highly stretched meshes. Hence, the implicit operator allows large time steps i.e. CFL numbers of the order of 1000. The proposed method is applied to three dimensional cases of viscous, turbulent flow around a wing, achieving dramatically improved convergence rates