On the Asymptotic Spectrum of Finite Element Matrix Sequences

We derive a new formula for the asymptotic eigenvalue distribution of stiffness matrices obtained by applying P1 Finite Elements with standard mesh refinement to the semi elliptic PDE of second order in divergence form −∇(K∇Tu) = f on Ω, u = g on ∂Ω. Here Ω ⊂ R and K is supposed to be piecewise continuous and pointwise symmetric semi positive definite. The symbol describing this asymptotic eigenvalue distribution depends on the PDE, but also both on the numerical scheme for approaching the underlying bilinear form and on the geometry of triangulation of the domain. Our work is motivated by recent results on the superlinear convergence behavior of the Conjugate Gradient method, which requires the knowledge of such asymptotic eigenvalue distributions for sequences of matrices depending on a discretization parameter h when h→ 0. We compare our findings with similar results for the Finite Difference method which were published over the last years. In particular we observe that also our sequence of stiffness matrices is part of the class of Generalized Locally Toeplitz sequences for which many theoretical tools are available. This enables us to derive some results on the conditioning and preconditioning of such stiffness matrices.