A parallel fictitious domain multigrid preconditioner for the solution of Poisson’s equation in complex geometries

A parallel multilevel preconditioner based on domain decomposition and fictitious domain methods has been presented for the solution of the Poisson equation in complicated geometries. Rectangular blocks with matching grids on interfaces on a structured rectangular mesh have been used for the decompo...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering Vol. 194; no. 45; pp. 4845 - 4860
Main Authors: Singh, K.M., Williams, J.J.R.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.11.2005
Elsevier
Subjects:
Computational techniques
Conjugate gradient
Domain decomposition
Exact sciences and technology
Fictitious domain method
Fundamental areas of phenomenology (including applications)
Heat conduction
Heat transfer
Mathematical methods in physics
Multigrid preconditioner
Parallel computing
Physics
Poisson equation
Poisson equation
Conjugate gradient
Multigrid preconditioner
Parallel computing
Fictitious domain method
Domain decomposition
Difference scheme
Turbulent flow
Interior point method
Conjugate gradient methods
Spheres
Parallel algorithms
Laplacian
Grids
Incompressible flow
Multigrid
Linear algebra
Curved surface
Incompressible fluid
Preconditioning
Heat transfer
Thermal conduction
ISSN:0045-7825, 1879-2138
Online Access:Get full text
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Summary:A parallel multilevel preconditioner based on domain decomposition and fictitious domain methods has been presented for the solution of the Poisson equation in complicated geometries. Rectangular blocks with matching grids on interfaces on a structured rectangular mesh have been used for the decomposition of the problem domain. Sloping sides or curved boundary surfaces are approximated using stepwise surfaces formed by the grid cells. A seven-point stencil based on the central difference scheme has been used for the discretization of the Laplacian for both interior and boundary grid points, and this results in a symmetric linear algebraic system for any type of boundary condition. The preconditioned conjugate gradient method has been used for the solution of this symmetric system. The multilevel preconditioner for the CG is based on a V-cycle multigrid applied to the Poisson equation on a fictitious domain formed by the union of the rectangular blocks used for the domain decomposition. Numerical results are presented for two typical Poisson problems in complicated geometries—one related to heat conduction, and the other one arising from the LES/DNS of incompressible turbulent flow over a packed array of spheres. These results clearly show the efficiency and robustness of the proposed approach.
ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2005.01.003