Robust BPX preconditioner for fractional Laplacians on bounded Lipschitz domains

Abstract

We propose and analyze a robust Bramble-Pasciak-Xu (BPX) preconditioner for the integral fractional Laplacian of order s ∈ ( 0 , 1 ) s\in (0,1) on bounded Lipschitz domains. Compared with the standard BPX preconditioner, an additional scaling factor 1 − γ ~ s 1-\widetilde {\gamma }^s , for some fixed γ ~ ∈ ( 0 , 1 ) \widetilde {\gamma } \in (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.