Talk by Zdeněk Strakoš

The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place this Wednesday (Nov 24, 2021) from 9:00 till 10:00 via Zoom virtually. The talk will be given by Zdeněk Strakoš through Zoom.

Speaker: Zdeněk Strakoš
Title: Numerical approximation of the spectrum of self-adjoint operators and operator preconditioning

Abstract: We consider operator preconditioning B^{-1}A, which is employed in the numerical solution of boundary value problems. Here, the self-adjoint operators A, B: H_0^1(Omega) to H^{-1}(Omega) are the standard integral/functional representations of the partial differential operators (-nabla cdot (k(x)nabla u)) and (-nabla cdot (g(x)nabla u)), respectively, and the scalar coefficient functions k(x) and g(x) are assumed to be continuous throughout the closure of the solution domain. The function g(x) is also assumed to be uniformly positive. When the discretized problem, with the preconditioned operator B_n^{-1}A_n, is solved with Krylov subspace methods, the convergence behavior depends on the distribution of the eigenvalues. Therefore it is crucial to understand how the eigenvalues of B_n^{-1}A_n are related to the spectrum of B^{-1}A. Following the path started in the two recent papers published in SIAM J. Numer. Anal. [57 (2019), pp. 1369-1394 and 58 (2020), pp. 2193-2211], the first part of the talk addresses the open question concerning the distribution of the eigenvalues of B_n^{-1}A_n formulated at the end of the second paper.

The second part generalizes some of the results to bounded and self-adjoint operators A, B: V -> V^{#}, where V^{#} denotes the dual of V. More specifically, provided that B is coercive and that the standard Galerkin discretization approximation properties hold, we prove that the whole spectrum of B^{-1}A: V -> V is approximated to an arbitrary accuracy by the eigenvalues of its finite dimensional discretization B_n^{-1}A_n.

The presented spectral approximation problem includes the continuous part of the spectrum and it differs from the eigenvalue problem studied in the classical PDE literature which addresses compact (solution) operators.

This is a joint work with Tomáš Gergelits and Bjørn Fredrik Nielsen.