A note on the asymptotic spectra of finite difference discretizations of second order elliptic partial differential equations
Stefano Serra‐Capizzano · Asian Journal of Mathematics · 2000
We consider Finite Difference discretizations of an elliptic second order PDE as -£\ ^7 laij(x)-rp-u(x) j = b(x) over (0, l) d with Dirichlet boundary conditions, where the dxd matrix A{x) = (aij(x)) is symmetric, uniformly positive definite and whose entries are Riemann integrable.We choose the discretization so that the resulting matrices {A n (A)} n form a sequence of Hermitian positive definite matrices.The eigenvalue distribution has been studied and characterized [23] in terms of weighted multidimensional Szego formulas.Here by using some tools introduced in a preceding paper [17] we analyze the spectral behaviour of the preconditioned matrix sequences {An 1 {B)An{A)}n so that B(x) is symmetric positive definite and with Riemann integrable entries.Some issues on efficient preconditioning strategies are discussed as well.