Chebyshev Smoothing With Adaptive Block‐FSAI Preconditioners for the Multilevel Solution of Higher‐Order Problems With the Partition of Unity Method

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 the efficient solution of linear systems arising from a conforming discretization of higher‐order partial differential equations via the partition of unity method (PUM). To this end, we consider the (anisotropic) biharmonic and triharmonic equations in two and three dimensions and discretize these problems with a ‐ or ‐regular PUM and polynomial exactness up to degree 5. We adapt existing adaptive algorithms for the construction of sparse approximate inverses to the native block structure of matrices arising in the PUM. Additionally, we also present a simplified formulation of the Chebyshev iteration of the fourth kind.