Polynomial of best uniform approximation to $x^{-1}$ and smoothing in two-level methods
Provides a theoretical foundation for a polynomial smoother in multigrid methods, ensuring robust convergence for scalar elliptic PDEs.
The authors derive a three-term recurrence for the polynomial of best uniform approximation to x^{-1} on a finite interval and apply it as a smoother in two-level methods for elliptic PDEs. They prove a smoothing property and achieve uniform convergence rates with respect to mesh parameters, coarsening ratio, and coefficient variation.
We derive a three-term recurrence relation for computing the polynomial of best approximation in the uniform norm to $x^{-1}$ on a finite interval with positive endpoints. As application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.