Matrix-Free Polynomial-Based Nonlinear Least Squares Optimized Preconditioning and Its Application to Discontinuous Galerkin Discretizations of the Euler Equations

We introduce a preconditioner that can be both constructed and applied using only the ability to apply the underlying operator. Such a preconditioner can be very attractive in scenarios where one has a highly efficient parallel code for applying the operator. Our method constructs a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show that this polynomial-based NLLS-optimized (PBNO) preconditioner significantly improves the performance of a discontinuous Galerkin (DG) compressible Euler equation model when run in an implicit-explicit time integration mode. The PBNO preconditioner achieves significant reduction in GMRES iteration counts and model wall-clock time, and significantly outperforms several existing types of generalized (linear) least squares (GLS) polynomial preconditioners. Comparisons of the ability of the PBNO preconditioner to improve DG model performance when employing the Stabilized Biconjugate Gradient algorithm (BICGS) and the basic Richardson (RICH) iteration are also included. In particular, we show that higher order PBNO preconditioning of the Richardson iteration (run in a dot product free mode) makes the algorithm competitive with GMRES and BICGS in a serial computing environment. Because the NLLS-based algorithm used to construct the PBNO preconditioner can handle both positive definite and complex spectra without any need for algorithm modification, we suggest that the PBNO preconditioner is, for certain types of problems, an attractive alternative to existing polynomial preconditioners based on linear least-squares methods.