Laplacian preconditioning of elliptic PDEs: A priori estimates of all the eigenvalues

Prof. Dr. Bjorn Fredrik Nielsen

Nov. 29, 2018, 12:30 p.m. HF 9904

The convergence properties of Krylov subspace methods applied to $Bx=b$ is often studied in terms of the condition number of the matrix $B$. This approach has many advantages, but the actual convergence speed of such algorithms may be influenced by the entire distribution of the eigenvalues of $B$. In fact, as has been pointed out by several mathematicians, the condition number can yield misleading information about which preconditioner should be employed.

In this talk we therefore investigate whether a priori estimates of all the eigenvalues of a particular preconditioned linear system is readily available. More specifically, we analyze the spectrum of $B=L^{-1} A$, where $L$ and $A$ are the stiffness matrices associated with the Laplace operator and general second order elliptic operators, respectively. We show that the coefficient function $k(x)$ of the elliptic PDE provides information about the whole spectrum of $L^{-1} A$: if $k(x)$ is continuous or has small jump discontinuities, then the nodal values of $k(x)$ yield accurate approximations of all the individual eigenvalues.