Abstract
Abstract We propose a multi-level type operator that can be used in the framework of operator (or Caldéron) preconditioning to construct uniform preconditioners for negative order operators discretized by piecewise polynomials on a family of possibly locally refined partitions. The cost of applying this multi-level operator scales linearly in the number of mesh cells. Therefore, it provides a uniform preconditioner that can be applied in linear complexity when used within the preconditioning framework from our earlier work [Uniform preconditioners for problems of negative order, Math. Comp. 89 (2020), 645–674].
Highlights
In this work, we construct a multi-level type preconditioner for operators of negative orders −2s ∈ [−2, 0] that can be applied in linear time and yields uniformly bounded condition numbers
We propose a multi-level type operator that can be used in the framework of operator preconditioning to construct uniform preconditioners for negative order operators discretized by piecewise polynomials on a family of possibly locally refined partitions
It provides a uniform preconditioner that can be applied in linear complexity when used within the preconditioning framework from our earlier work [Uniform preconditioners for problems of negative order, Math
Summary
We construct a multi-level type preconditioner for operators of negative orders −2s ∈ [−2, 0] that can be applied in linear time and yields uniformly bounded condition numbers. We construct a suitable multi-level type operator BST that can be applied in linear complexity. For this construction, we require T to be a family of conforming partitions created by newest vertex bisection [7, 13]. In the aforementioned setting of having an arbitrary s ∈ [0, 1], this multi-level operator BST induces a uniform preconditioner GT, i.e., GT AT is uniformly well-conditioned, where the cost of applying GT scales linearly in dim VT.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
