A polynomial-based nonlinear least squares optimized preconditioner for continuous and discontinuous element-based discretizations of the Euler equations
Abstract
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 im- proves the performance of 2-D continuous Galerkin (CG) and
discontinuous Galerkin (DG) fluid dynamical research 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.
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.Collections
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