Multilevel preconditioning of graph-Laplacians: Polynomial approximation of the pivot blocks inverses

We consider the discrete system resulting from mixed finite element approximation of a second-order elliptic boundary value problem with Crouzeix–Raviart non-conforming elements for the vector valued unknown function and piece-wise constants for the scalar valued unknown function. Since the mass matrix corresponding to the vector valued variables is diagonal, these unknowns can be eliminated exactly. Thus, the problem of designing an efficient algorithm for the solution of the resulting algebraic system is reduced to one of constructing an efficient algorithm for a system whose matrix is a graph-Laplacian (or weighted graph-Laplacian).