University of Texas at Austin

Past Event: Babuška Forum

Separation of Variables with Non-Self Adjoint Operators with Applications to Analysis of Waveguides

Professor Leszek F. Demkowicz, Oden Institute for Computational Engineering

10 – 11AM
Friday Mar 22, 2024

POB 6.304 & Zoom

Abstract

I have been teaching separation of variables for over 30 years, and I have always been instructing students to look for a self-adjoint problem first. We know then that the separation constant (the eigenvalue) is real and, if the operator happens to be positive definite, we also know that the constant is positive. This a-priori knowledge about the separation constant simplifies greatly the solution of the problem. More importantly, by the Sturm-Liouville theory (Spectral Theorem for Self-Adjoint Operators), we know that the eigenvectors form simultaneously a basis for the L2 as well H1 space, which opens up the way for a rigorous well-posedness proof.

Only recently, when studying acoustical and electromagnetic waveguides, I have run into situations when I have to perform the separation of variables with a non-self adjoint operator. This has led me to a half-year long study of excellent and fundamental book by Gohberg and Krein [1] and a number of fundamental results that I have learned (and should have known a long time ago).

In the talk, I will illustrate the deep theory for non-self adjoint operators with a simple model acoustic waveguide problem with impedance boundary conditions.

[1] I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space (Translations of Mathematical Monographs). American Mathematical Society, 1965 (Russian edition), vol. 18.

Biography

Leszek F. Demkowicz is Assistant Director of the Oden Institute for Computational Engineering and Sciences and holder of W. A. ``Tex'' Moncrief, Jr. Chair in Computational Engineering and Sciences II. He is a Professor in the Departments of Aerospace Engineering and Engineering Mechanics, and Mathematics, at the University of Texas at Austin. He has a M.S. in mathematics from Jagiellonian University, and M.S., Ph.D. and Sc.D. degrees in mechanics from Cracow University of Technology (CUT). Dr. Demkowicz authored and/or co-authored several books, over 200 journal articles, book chapters and technical reports in the general area of computational mathematics and mechanics. His work and scientific interests span across numerical analysis, adaptive finite element methods, wave propagation problems, including acoustics, elastodynamics and electromagnetics, and CFD.

Separation of Variables with Non-Self Adjoint Operators with Applications to Analysis of Waveguides

Event information

Date
10 – 11AM
Friday Mar 22, 2024
Location POB 6.304 & Zoom
Hosted by Blake Christierson