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 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.