A cost-effective multigrid projection operator

A novel heuristic residual analysis is proposed to derive a computationally cost-effective residual projection operator in multigrid with the five-point Red-Black Gauss-Seidel relaxation for solving the two-dimensional Poisson equation. This optimal residual injection operator is as cheap as the trivial injection operator, but is more efficient than the costly full-weighting operator and achieves near-optimal convergence rate.