On additive Schwarz preconditioners for sparse grid discretizations

Based on the framework of subspace splitting and the additive Schwarz scheme, we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems. For a BXP-like preconditioner we derive an estimate of the optimal orderO(1) and for a HB-like variant we obtain an estimate of the orderO(k 2 ·2 k/2 ), wherek denotes the number of levels employed. Furthermore, we confirm these results by numerically computed condition numbers.