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A finite-element coarse-grid projection method for incompressible flow simulations

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Abstract

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure-correction schemes used for the incompressible Navier-Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. The minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions.

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Acknowledgments

AK wishes to thank Dr. Michael Lentine, Dr. Peter Minev, Dr. Saad Ragab, and Dr. Omer San for helpful discussions. Moreover, AK would like to thank the reviewers for their insightful comments.

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Authors and Affiliations

  1. Institute for Computational & Mathematical Engineering, Department of Geophysics, Stanford University, Stanford, CA, 94305, USA

    Ali Kashefi

  2. Engineering Science and Mechanics Program, Department of Biomedical Engineering and Mechanics, Virginia Tech, Blacksburg, VA, 24061, USA

    Ali Kashefi & Anne E. Staples

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  1. Ali Kashefi
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  2. Anne E. Staples
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Correspondence to Ali Kashefi.

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Communicated by: Axel Voigt

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Kashefi, A., Staples, A.E. A finite-element coarse-grid projection method for incompressible flow simulations. Adv Comput Math 44, 1063–1090 (2018). https://doi.org/10.1007/s10444-017-9573-5

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