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An Introduction to Element-Based Galerkin Methods on Tensor-Product Bases

Analysis, Algorithms, and Applications

  • Textbook
  • © 2020


  • The construction of element matrices and the resulting matrices are shown for all the differential operators discussed. This helps the reader understand the material clearly and assists them in building their numerical algorithms
  • Both modal and nodal basis functions are discussed throughout the text, including examples for the continuous and discontinuous Galerkin method, it is shown how to combine both of these methods into one piece of code; moreover, a description of hybridized discontinuous Galerkin method is included in this textbook. A sample solution along with the order of accuracy and time-to-solution (work-precision diagrams) are shown for a variety of test problems for different types of equations (e.g., hyperbolic and elliptic)
  • Sample code for student projects are provided on my Github page with code in both matlab and Julia (see

Part of the book series: Texts in Computational Science and Engineering (TCSE, volume 24)

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About this book

This book introduces the reader to solving partial differential equations (PDEs) numerically using element-based Galerkin methods. Although it draws on a solid theoretical foundation (e.g. the theory of interpolation, numerical integration, and function spaces), the book’s main focus is on how to build the method, what the resulting matrices look like, and how to write algorithms for coding Galerkin methods. In addition, the spotlight is on tensor-product bases, which means that only line elements (in one dimension), quadrilateral elements (in two dimensions), and cubes (in three dimensions) are considered. The types of Galerkin methods covered are: continuous Galerkin methods (i.e., finite/spectral elements), discontinuous Galerkin methods, and hybridized discontinuous Galerkin methods using both nodal and modal basis functions. In addition, examples are included (which can also serve as student projects) for solving hyperbolic and elliptic partial differential equations, includingboth scalar PDEs and systems of equations.

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Table of contents (21 chapters)

  1. Introduction

  2. One-Dimensional Problems

  3. Multi-Dimensional Problems

Authors and Affiliations

  • Department of Applied Mathematics, Naval Postgraduate School, Monterey, USA

    Francis X. Giraldo

About the author

Francis (Frank) Giraldo is a Distinguished Professor of Applied Mathematics at the Naval Postgraduate School and a founding member of the Scientific Computing group. He and his team built the NUMA model using the element-based Galerkin (EBG) methods described in this text; NUMA is a Navier-Stokes solver used for atmospheric, ocean, and fluid dynamics simulations. Frank Giraldo (and colleagues) hosted the 2012 Gene Golub SIAM Summer School on Simulation and Supercomputing in the Geosciences where EBG methods was one of the topics of the summer course. In addition, Frank has served on the National Earth Systems Prediction Capability working groups for over 10 years, and has served on the Department of Energy’s INCITE panels for over 5 years (including chairing the committee a number of times).

Bibliographic Information

  • Book Title: An Introduction to Element-Based Galerkin Methods on Tensor-Product Bases

  • Book Subtitle: Analysis, Algorithms, and Applications

  • Authors: Francis X. Giraldo

  • Series Title: Texts in Computational Science and Engineering

  • DOI:

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

  • Hardcover ISBN: 978-3-030-55068-4Published: 31 October 2020

  • Softcover ISBN: 978-3-030-55071-4Published: 01 November 2021

  • eBook ISBN: 978-3-030-55069-1Published: 30 October 2020

  • Series ISSN: 1611-0994

  • Series E-ISSN: 2197-179X

  • Edition Number: 1

  • Number of Pages: XXVI, 559

  • Number of Illustrations: 3 b/w illustrations, 168 illustrations in colour

  • Topics: Partial Differential Equations, Computational Science and Engineering, Numerical Analysis

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