### Abstract

This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.

Language | English |
---|---|

Pages | 287-304 |

Number of pages | 18 |

Journal | International Journal of Computational Fluid Dynamics |

Volume | 14 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jan 2001 |

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### Keywords

- domain decomposition method
- finite elements
- finite volumes
- multigrid algorithm
- Navier-Stokes equations
- parallel computing
- triangular meshes

### Cite this

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*International Journal of Computational Fluid Dynamics*, vol. 14, no. 4, pp. 287-304. https://doi.org/10.1080/10618560108940730

**A hybrid domain decomposition and multigrid method for the acceleration of compressible viscous flow calculations on unstructured triangular meshes.** / Dolean, Victorita; Lanteri, Stéphane.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A hybrid domain decomposition and multigrid method for the acceleration of compressible viscous flow calculations on unstructured triangular meshes

AU - Dolean, Victorita

AU - Lanteri, Stéphane

PY - 2001/1/1

Y1 - 2001/1/1

N2 - This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.

AB - This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.

KW - domain decomposition method

KW - finite elements

KW - finite volumes

KW - multigrid algorithm

KW - Navier-Stokes equations

KW - parallel computing

KW - triangular meshes

UR - http://www.scopus.com/inward/record.url?scp=0035634180&partnerID=8YFLogxK

U2 - 10.1080/10618560108940730

DO - 10.1080/10618560108940730

M3 - Article

VL - 14

SP - 287

EP - 304

JO - International Journal of Computational Fluid Dynamics

T2 - International Journal of Computational Fluid Dynamics

JF - International Journal of Computational Fluid Dynamics

SN - 1061-8562

IS - 4

ER -