Optimal Injection Operator and High Order Schemes
for Multigrid Solution of 3D Poisson Equation

Jules Kouatchou
School of Engineering
Morgan State University
5200 Perring Parkway
Baltimore, MD 21239, USA

Jun Zhang
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA


We present a multigrid solution of the three dimensional Poisson equation with a fourth order 19-point compact finite difference scheme. Using a red-black ordering of the grid points and some geometric considerations, we derive an optimal scaled injection operator for the multigrid algorithm. Numerical computations show that this operator yields not only the smallest overall CPU time, but also the best convergence rate compared to other more traditional projection operators. In addition, we present a family of 19-point compact schemes and numerically show that each one has a different optimal scaled injection operator.

Key words: 3D Poisson equation, fourth order compact discretization, multigrid method, scaled injection operator.

1991 Mathematical Subject Classification : 65F10, 65N06, 65N22, 65N55.

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This paper has been published in International Journal of Computer Mathematics, Vol. 76, 173-190 (2000),