Multilevel Splitting of Weighted Graph-Laplacian Arising in Non-conforming Mixed FEM Elliptic Problems

Abstract

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. In this paper we study the construction of optimal order preconditioners based on algebraic multilevel iterations (AMLI). The weighted graph-Laplacian for the model 2-D problem is considered. We assume that the finest triangulation is obtained after recursive uniform refinement of a given coarse mesh. The introduced hierarchical splitting is the first important contribution of this article. The proposed construction allows for a local analysis of the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality. This is an important characteristic of the splitting and is associated with the angle between the two hierarchical FEM subspaces. The estimates of the convergence rate and the computational cost at each iteration show that the related AMLI algorithm with acceleration polynomial of degree two or three is of optimal complexity.