Algebraic Multigrid for Moderate Order Finite Elements

The paper discusses algebraic multigrid (AMG) methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with Lagrangian finite elements of order at most 4. The resulting system matrices do not have the M-matrix property that is used by standard analyzes of classical AMG and aggregation-based AMG methods. A unified approach is presented that allows to extend these analysis. It uses an intermediate M-matrix and highlights the role of the spectral equivalent constant that relate this matrix to the original system matrix. This constant is shown to be bounded independently of the problem size and jumps in the coefficients of the partial differential equations that are located at elements’ boundaries. For two dimensional problems, it is further shown to be uniformly bounded if the angles in the triangulation also satisfy a uniform bound. Because the intermediate M-matrix can be computed automatically, an alternative strategy is also investigated that defines the AMG preconditioners from this matrix instead of from the original matrix. Numerical experiments are presented that asses both strategies using publicly available state-of-the-art implementations of classical AMG and aggregation-based AMG methods.