Multigrid methods: Grid transfer operators and subdivision schemes

The convergence rate of a multigrid method depends on the properties of the smoother and the so-called grid transfer operator. In this paper we define and analyze new grid transfer operators with a generic cutting size which are applicable for high order problems. We enlarge the class of available geometric grid transfer operators by relating the symbol analysis of the coarse grid correction with the approximation properties of univariate subdivision schemes. We show that the polynomial generation property and stability of a subdivision scheme are crucial for convergence and optimality of the corresponding multigrid method. We construct a new class of grid transfer operators from univariate primal binary and ternary pseudo-spline symbols. Our numerical results illustrate the behavior of the new grid transfer operators and provide promising preliminary results for the bivariate case.