University of Texas at Austin


Etching Progress into Memory: Leszek Demkowicz’ 'Mathematical Theory of Finite Elements’ Published by SIAM

By Aira Balasubramanian

Published Jan. 9, 2024

Leszek Demkowicz

After being approached by thirteen students eager to explore this fundamental concept in 2002, Leszek Demkowicz has taught ‘Mathematical Theory of Finite Elements’ at the Oden Institute for Computational Engineering and Sciences over thirteen times. Writing and rewriting this course over the years refined and developed his lecture notes into a highly original body of work, uniquely designed to guide students through the material while becoming inexorably tied to his own research interests.

Publishing and preserving this material in print became a priority to Demkowicz. Hot off the Society of Industrial and Applied Mathematics (SIAM)’s presses, ‘The Mathematical Theory of Finite Elements’ was published in September 2023, marking revolutionary progress in supporting graduate students in computational sciences and engineering programs at the Oden Institute and beyond.

In an efficient 200 pages, the book focuses on theories of discrete stability and conforming elements forming the exact sequence. Demkowicz covers Ritz-Galerkin methods for physical systems, Mikhiln’s theory, the Lax-Milgram and Cea’s lemma, the Babuska theorem, and Brezzi’s Theory, organized chronologically through these methods’ development. He concludes by introducing discontinuous Petrov-Galerkin (DPG) methods with optimal test functions, an award winning method he co-invented in 2009 alongside Jay Gopalakrishnan, which has formed the focus of his research work.

This work is Demkowicz’s seventh published book, joining an array of textbooks covering Hp Adaptive finite elements, Dimensional Epileptic and Maxwell problems, and Applied Functional Analysis, which underscore his academic and industrial expertise in applied mathematics, continuum mechanics, and computational methods for partial differential equations, with focus on finite element and boundary element methods. Serving as the Assistant Director of the Oden Institute and lead of the Electromagnetics and Acoustics Group within the Institute, he also holds the W. A ‘Tex’ Moncrief Chair in Computational Engineering and Sciences. Demkowicz's impact on student success is unparalleled through his writing and research.