Public Lecture of the City of Poznań within the programme "Academic and Scientific Poznań"
Discrete Stability - The Perpetual Challenge in Numerical Approximation of Partial Differential Equations
Prof. Leszek Demkowicz, Oden Institute for Computational Science and Engineering, UT at Austin
June 13, 2022 at 13.30, Poznan University of Technology, Campus Warta, building CW, room 123 BT
The lecture will be held in the hybrid mode:
at the lecture room 123 BT, Campus Warta, Poznan University of Technology, and through a continuous live stream via the PUT YouTube Channel
and the specific link to the lecture will be generated just before its beginning there.
Computer simulations provide an indispensable foundation for the majority of complex engineering and science projects. They have long provided a basis for the analysis and design of buildings, bridges, cars, airplanes, ships and submarines, oil exploration. They are being used to investigate and design new drugs, materials, optical fibers, production lines and even diapers.
Most of the models are described in terms of systems of Partial Differential Equations (PDEs). The PDEs are then discretized, i.e., approximated with resulting systems of linear and nonlinear algebraic equations that can be solved on the computer. Finite Element (FE) and Finite Difference (FD) methods are the most popular techniques of discretization.
The discretization process is subjected to a careful verification including a convergence analysis; when the element (mesh) size goes to zero, the corresponding FE or FD solution must converge to the exact solution of the PDE system. In my 46-year career on both sides of the ocean, I have been involved with the design of new FE discretization schemes and their application to solve practical problems but also with providing theoretical foundations for such methods.
In the lecture, I will take you through the historical development of the Galerkin and FE method which, incidentally, coincides with my personal growth. Starting with Ritz and Galerkin, we will go through the fundamental results of Mikhlin, Cea and Babuska. The underlying concept is that of DISCRETE STABILITY and it relates to the fundamental Closed Range Theorem of the greatest Polish mathematician - Stefan Banach. I hope to communicate the main concepts to a very general audience.