Multigrid methods: grid transfer operators and subdivision schemes
For researchers in numerical analysis and scientific computing, this work provides a principled way to design grid transfer operators, though it is an incremental extension of existing theory.
The paper introduces new grid transfer operators for multigrid methods, derived from subdivision schemes, and demonstrates their effectiveness for high-order problems. Numerical results confirm improved convergence properties.
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 primal binary and ternary pseudo-spline symbols. Our numerical results illustrate the behavior of the new grid transfer operators.