On the Asymptotic Spectrum of Finite Element Matrix Sequences

Abstract

We derive a new formula for the asymptotic eigenvalue distribution of stiffness matrices obtained by applying P 1 finite elements with standard mesh refinement to the semielliptic PDE of second order in divergence form −∇(K∇ T u) = f on Ω, u = g on ∂Ω.Here Ω ⊂ R 2 , and K is supposed to be piecewise continuous and pointwise symmetric semipositive 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 in recent years. In particular we observe that 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.

AMG preconditioning for nonlinear degenerate parabolic equations on nonuniform grids with application to monument degradation

“…Such information consists in the locally Toeplitz structure used and in the related spectral features: conditioning, subspaces related to small eigenvalues, global spectral behaviour, etc. (see [17] and [2]). …”

“…Of course, as in the onedimensional case the use of Dirichlet boundary conditions reduces the gridding to the N 2 internal points; also in this case other boundary conditions can be considered in a similar way. With this choice, the grid spacing is h [1] k = x k − x k−1 and h [2] k = y k − y k−1 . The finite difference discretization of the differential operator (2.1) can be generalised as follows:…”

“…When needed, appropriate finite rank corrections for boundary conditions are also included in the first/last blocks of A and first/last rows of the blocks in B. Notice that the pattern of the matrix L (2) D(u) reminds a tensor structure that unfortunately cannot be formalised exactly if the function D(u) is not separable, due to the varying coefficients in the underlying PDE (for the spectral analysis of such structures refer to [14] and references therein).…”

“…Collecting the unknowns u i, j in a vector u using the lexicographical ordering, we denote by L (2) D(u) the matrix representing the differential operator. This matrix has the same sparsity pattern as the tensor product L (1)…”

“…We consider the domain Ω = [a 1 , b 1 ] × [a 2 , b 2 ] ⊂ R 2 and the grid points x i, j = (g [1] (i/(N + 1)), g [2] ( j/(N + 1))), where…”