Algebraic Multilevel Preconditioning of Finite Element Matrices Based on Element Agglomeration

We consider an algebraic multilevel preconditioning method for SPD matrices resulting from finite element discretization of elliptic PDEs. In particular, we focus on non-M matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The coarse-grid element matrices are simply Schur complements computed from local neighborhood matrices (agglomerate matrices), i.e., small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation. In addition, performing the elimination of fine-degrees of freedom locally, but then without neglecting any fill-in, offers the opportunity to construct a new kind of incomplete LU factorization of the pivot matrix at every level. Based on these components an algebraic multilevel preconditioner is defined. The method can also be applied to systems of PDEs. A numerical analysis shows its efficiency and robustness.