Optimal injection operator and high order schemes for multigrid solution of 3D Poisson equation

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.