An implicit block ILU smoother for preconditioning of Newton-Krylov solvers with application in high-order stabilized finite-element methods.

*(English)*Zbl 1441.76052Summary: This paper presents an efficient and highly-parallelizable preconditioning technique for Newton-Krylov solvers. The proposed method can be viewed as a generalization of the implicit line smoothing technique by extending the groups of implicitly-solved unknowns from lines to blocks. The blocks are formed by partitioning the computational domain such that the strong connections between unknowns are not broken by the partition boundaries. The ILU algorithm is used to obtain an approximate (or exact) factorization for each block. Then, a block-Jacobi iteration is formulated in which the degrees of freedom within the blocks are solved implicitly. To stabilize the iterations for high-CFL systems, a dual-CFL strategy, with a lower CFL in the preconditioner matrix, is developed. The performance of the proposed method as a linear preconditioner is demonstrated for second- and third-order steady-state solutions of Reynolds-Averaged Navier-Stokes (RANS) equations on the NASA Common Research Model (CRM), including the high-lift configuration. For the studied test cases, it is shown that in comparison with the traditional ILU(k) method, the proposed preconditioner requires significantly less memory, and it can result in notably faster solutions.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

##### Keywords:

computational fluid dynamics; Newton-Krylov methods; steady- state RANS simulations; preconditioning techniques; high-order methods; stabilized finite-element methods
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\textit{B. R. Ahrabi} and \textit{D. J. Mavriplis}, Comput. Methods Appl. Mech. Eng. 358, Article ID 112637, 29 p. (2020; Zbl 1441.76052)

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