Estimates for generalized sparse grid hierarchical basis preconditioners

Abstract

We reconsider some estimates from the 1994 paper [6] concerning the hierarchical basis preconditioner for sparse grid discretizations. The improvement is in three directions: We consider arbitrary space dimensions <em>d</em> > 1, give bounds for generalized sparse grid spaces with arbitrary monotone index set Λ, and show that the bounds are sharp up to constants depending only on <em>d</em>, at least for a subclass of Λ containing full grid, standard sparse grid spaces, and energy-norm optimized sparse grid spaces.