Comparison of second- and fourth-order discretizations for multigrid Poisson solvers (Q1357336)

The authors compare multigrid algorithms for solving the Poisson equation discretized by using the five-point stencil (approximation order \(h^2\)) and a nine-point stencil (approximation order \(h^4\)). Within the multigrid algorithm different projection operators (e.g. full-injection, half-injection, full-weighting) and Gauss-Seidel smoothers with different orderings (red-black, four-colour, lexicographical) are used. The cost of arithmetical work and the storage cost of the multigrid \(V\)-cycle are discussed briefly for both discretizations. The presented numerical results show that the multigrid algorithm with full-weighting and red-black Gauss-Seidel smoother is the fastest algorithm in the case of the nine-point discretization. Furthermore, the experiments performed on serial and vector machines illustrate that the nine-point formula is superior to the five-point formula in both accuracy and computational efficiency.