Chebyshev smoothing with adaptive block-FSAI preconditioners for the multilevel solution of higher-order problems

Abstract: In this paper, we assess the performance of adaptive and nested factorized sparse approximate inverses as smoothers in multilevel V-cycles, when smoothing is performed following the Chebyshev iteration of the fourth kind. For our test problems, we rely on the partition of unity method to discretize the biharmonic and triharmonic equations in a multilevel manner. Inspired by existing algorithms, we introduce a new adaptive algorithm for the construction of sparse approximate inverses, based on the block structure of matrices arising in the partition of unity method. Additionally, we also present a new (and arguably simpler) formulation of the Chebyshev iteration of the fourth kind.