Finite Elements I: Approximation and Interpolation

This book is a comprehensive textbook suitable for graduate coursework, professional engineering, and academic research. It covers the fundamental ideas of finite elements, including Lagrange, vector-valued, and quasi-interpolation operators, and local commuting projections. It also includes chapters on functional analysis and implementation aspects. The text is divided into short chapters and includes exercises and solutions.

Format: Paperback / softback
Length: 325 pages
Publication date: 20 February 2022
Publisher: Springer Nature Switzerland AG

This comprehensive textbook serves as the first installment of a three-part series designed for advanced study, professional engineering practice, and academic research. It is also well-suited for graduate flipped courses. Each volume is thoughtfully organized into concise chapters, making it convenient for teaching purposes. Each chapter can be covered in a single teaching session, and it includes comprehensive exercises and solutions accessible through a dedicated website. The key concepts are introduced during lectures, with the remaining content designated as reading material to foster a deeper understanding. To engage the reader, the text seamlessly blends examples, fundamental principles, rigorous proofs, and references to the literature, thereby enhancing scientific literacy.

Volume I is structured into 23 chapters, along with two appendices covering Banach and Hilbert spaces and differential calculus. This volume primarily delves into the foundational concepts related to the construction of finite elements and their approximation capabilities. It explores both all-purpose Lagrange finite elements and vector-valued finite elements, which are essential for accurately approximating the divergence and curl operators. Additionally, it presents and analyzes quasi-interpolation operators and local commuting projections. The book begins with four chapters on functional analysis, providing a wealth of examples and counter-examples to familiarize the reader with fundamental facts on Lebesgue integration and weak derivatives. Volume I also provides a review of important implementation considerations when developing or utilizing a finite element toolbox, such as mesh orientation and the enumeration of degrees of freedom.

Weight: 584g
Dimension: 154 x 234 x 20 (mm)
ISBN-13: 9783030563424
Edition number: 1st ed. 2021