A Class of Efficient Locally Constructed Preconditioners Based on Coarse Spaces
2019
In this paper we present a class of robust and fully algebraic two-level preconditioners
for SPD matrices. We introduce the notion of algebraic local SPSD splitting of an SPD matrix
and we give a characterization of this splitting. This splitting leads to construct
algebraically and locally a class of efficient coarse spaces which bound the spectral condition number of the
preconditioned system by a number defined a priori. We also introduce the τ-filtering subspace.
This concept helps compare the dimension minimality of coarse spaces. Some PDEs-dependant
preconditioners correspond to a special case. The examples of the algebraic coarse spaces in this
paper are not practical due to expensive construction. We propose a heuristic approximation
that is not costly. Numerical experiments illustrate the efficiency of the proposed method.
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