Preconditioning harmonic unsteady potential flow calculations

This paper considers finite element discretisations of the Helmholtz equation and its generalisation arising from harmonic acoustics perturbations to a non-uniform 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 1D 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 2D Helmholtz equation, grid independent convergence is achieved using a QMR Krylov solver, significantly outperforming the popular SSOR preconditioner. Impressive results are also presented on more complex domains, including an axisymmetric aircraft engine inlet with non-stagnant mean flow and modal boundary conditions

Fast iterative solvers for convection-diffusion control problems

In this manuscript, we describe effective solvers for the optimal control of stabilized convection-diffusion problems. We employ the local projection stabilization, which we show to give the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the �first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to demonstrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the mesh size h, and the regularization parameter β, for a range of problems

Some Preconditioning Techniques for Saddle Point Problems

Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud \ud The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336