A closed-form multigrid smoothing factor for an additive Vanka-type smoother applied to the Poisson equation

We consider an additive Vanka-type smoother for the Poisson equation discretized by the standard finite difference centered scheme. Using local Fourier analysis, we derive analytical formulas for the optimal smoothing factors for vertex-wise and element-wise Vanka smoothers. In one dimension the element-wise Vanka smoother is equivalent to the scaled mass operator obtained from the linear finite element method and in two dimensions the element-wise Vanka smoother is equivalent to the scaled mass operator discretized by bilinear finite element method plus a scaled identity operator. Based on these findings, the mass matrix obtained from finite element method can be used as a smoother for the Poisson equation, and the resulting mass-based relaxation scheme yields small smoothing factors in one, two, and three dimensions, while avoiding the need to compute an inverse of a matrix. Our analysis may help better understand the smoothing properties of additive Vanka approaches and develop fast solvers for numerical solutions of other partial differential equations.