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We consider a second-order elliptic problem in mixed form that has to be
solved as a part of a projection algorithm for unsteady Navier-Stokes
equations. The use of Crouzeix-Raviart non-conforming elements for the
velocities and piece-wise constants for the pressure provides a locally
mass-conservative algorithm. Then, the Crouzeix-Raviart mass matrix is
diagonal, and the velocity unknowns can be eliminated exactly. The reduced
matrix for the pressure is referred to as weighted graph-Laplacian.
Construction of optimal order preconditioners based on algebraic multilevel iterations (AMLI) is considered. We define the hierarchical two-level transformations and corresponding 2x2 splittings locally for macroelements associated with the edges of the coarse triangulation, and derive estimates for the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality. We discuss algorithmic and computational aspects of the two- and multilevel methods and present results from performed numerical experiments. |