ROBUST BPX PRECONDITIONER FOR FRACTIONAL LAPLACIANS ON BOUNDED LIPSCHITZ DOMAINS

We propose and analyze a robust Bramble-Pasciak-Xu (BPX) pre-conditioner for the integral fractional Laplacian of order s is an element of(0,1) on bounded Lipschitz domains. Compared with the standard BPX preconditioner, an additional scaling factor 1 - (gamma) over tilde (s), for some fixed (gamma) over tilde is an element of(0, 1), is incorporated to the coarse levels. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain uniformly bounded with respect to both the number of levels and the fractional power.

Automated summary

We propose and analyze a robust Bramble-Pasciak-Xu (BPX) pre-conditioner for the integral fractional Laplacian on bounded Lipschitz domains. The additional scaling factor incorporated to the coarse levels ensures uniformly bounded condition numbers for quasi-uniform or graded bisection grids.