Multigrid defect correction and fourth-order compact scheme for Poisson's equation

This paper presents an analysis of a multigrid defect correction to solve a fourth-order compact scheme discretization of the Poisson's equation. We focus on the formulation, which arises in the velocity/pressure decoupling methods encountered in computational fluid dynamics. Especially, the Poisson's equation results of the divergence/gradient formulation and Neumann boundary conditions are prescribed. The convergence rate of a multigrid defect correction is investigated by means of an eigenvalues analysis of the iteration matrix. The stability and the mesh-independency are demonstrated. An improvement of the convergence rate is suggested by introducing the damped Jacobi and Incomplete Lower Upper smoothers. Based on an eigenvalues analysis, the optimal damping parameter is proposed for each smoother. Numerical experiments confirm the findings of this analysis for periodic domain and uniform meshes which are the working assumptions. Further numerical investigations allow us to extend the results of the eigenvalues analysis to Neumann boundary conditions and non-uniform meshes. The Hodge-Helmholtz decomposition of a vector field is carried out to illustrate the computational efficiency, especially by making comparisons with a second-order discretization of the Poisson's equation solved with a state of art of algebraic multigrid method.