Block triangular preconditioners for linearization schemes of the Rayleigh–Bénard convection problem

Abstract

Summary In this paper, we compare two block triangular preconditioners for different linearizations of the Rayleigh–Bénard convection problem discretized with finite element methods. The two preconditioners differ in the nested or nonnested use of a certain approximation of the Schur complement associated to the Navier–Stokes block. First, bounds on the generalized eigenvalues are obtained for the preconditioned systems linearized with both Picard and Newton methods. Then, the performance of the proposed preconditioners is studied in terms of computational time. This investigation reveals some inconsistencies in the literature that are hereby discussed. We observe that the nonnested preconditioner works best both for the Picard and for the Newton cases. Therefore, we further investigate its performance by extending its application to a mixed Picard–Newton scheme. Numerical results of two‐ and three‐dimensional cases show that the convergence is robust with respect to the mesh size. We also give a characterization of the performance of the various preconditioned linearization schemes in terms of the Rayleigh number.

Analysis of Failure Models and Deformation Evolution Process of Geological Hazards in Ganzhou City, China

“…For example, from 1998 to July 2019, there were 22,377 geohazard incidents in China's Southern Jiangxi Province. In 2015, 2,196 landslide events that caused significant casualties and economic damage were reported (Ke et al, 2017). The southern Jiangxi region has a complex geological structure and bedrock lithology, diverse topography and geomorphology, widespread engineering works, and experiences heavy rainfall.…”

A field‐split preconditioning technique for fluid‐structure interaction problems with applications in biomechanics

“…22,23 This is the reason why the FS preconditioner is a combination of the physics-based and domain decomposition approaches. Preconditioners of FS type have been used by the authors on Rayleigh-Bénard convection problems, [24][25][26] and the intention of this paper is to investigate their capabilities for biomedical FSI problems.…”

“…22,23 This is the reason why the FS preconditioner is a combination of the physics-based and domain decomposition approaches. Preconditioners of FS type have been used by the authors on Rayleigh-Bénard convection problems, [24][25][26] and the intention of this paper is to investigate their capabilities for biomedical FSI problems.The GMG is chosen over the AMG because a monolithic Arbitrary Lagrangian-Eulerian (ALE) formulation is used for the FSI problem, where we exploit the high accuracy of the GMG intergrid operators in preserving at each level the same fluid-solid decomposition as at the finest level.ALE schemes have been widely used to simulate biomedical FSI problems. 27-29 A standard ALE approach moves the mesh to follow the elastic body movements.…”

Construction of H-Refined Continuous Finite Element Spaces with Arbitrary Hanging Node Configurations and Applications to Multigrid Algorithms

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“…This corresponds to a Newton-Raphson iterative scheme [24]. Such a scheme is used in the last example where a Navier-Stokes flow is considered.…”