Polynomial-Based Nonlinear Least Squares Optimized Preconditioning and its Application to Continuous and Discontinuous Element-Based Discretizations of the Euler Equations

We introduce a method for constructing 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 2-D continuous Galerkin (CG) and discontinuous Galerkin (DG) compressible Euler equation models when run in an implicit-explicit time integration mode. When employed in a serially computed Schur-complement form of the 2-D CG model with positive definite spectrum, the PBNO preconditioner achieves greater reductions in GMRES iterations and model wall-clock time compared to the analogous linear least-squares-derived Chebyshev polynomial preconditioner. Whereas constructing a Chebyshev preconditioner to handle the complex spectrum of the DG model would introduce an element of arbitrariness in selecting the appropriate convex hull, construction of a PBNO preconditioner for the 2-D DG model utilizes precisely the same objective NLLS algorithm as for the CG model. As in the CG model, the PBNO preconditioner achieves significant reduction in GMRES iteration counts and model wall-clock time. Comparisons of the ability of the PBNO preconditioner to improve CG and 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 (which is run in a dot product free mode) makes the algorithm competitive with GMRES and BICGS in a serial computing environment, especially when employed in a DG model. 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.