An Introduction to the Finite Element Method for Differential Equations. Mohammad Asadzadeh
and partial differential equations, their classifications, well‐posedness (as the proof of Reisz and Lax–Milgram Theorems) and formulation of the corresponding initial‐ and initial‐boundary value problems. The concept of fundamental solutions using Green's functions approaches are discussed and necessary mathematical tools and environment are introduced in some details.
(II) One‐space dimensional problems: Chapters 3–7 concern the polynomial approximations, polynomial interpolation, quadrature rules (numerical integration), iterative numerical methods to solve linear system of equations and finite element procedure for the one‐space dimensional boundary value problems (BVPs), initial value problems (IVPs), and initial boundary value problems (IBVPs).
(III) Problems in higher (
Whole or selected parts of the book is suitable as course material and for particular purposes. As outlined in table below, I have used some minor parts of Chapters 1 and 2 and the whole Chapters 3, and 5–7, together with a theory and a computer assignment, for a 7.5 credit points for undergraduates at Chalmers University of Technology. The whole book: Chapters 1–3 (a less pronounced cover of Chapters 5–7) together with Chapters 8–10 has been the course material for upper undergraduates and some graduates at Chalmers for more than two decades (since 1995). This part is associated with two, somewhat involved, theory and computer assignments and generates the same credit points 7.5, for a somewhat higher‐level audience. There is a, 5 credit points, intensive course on Chapters 2–3 and 8–10 for some graduates. (According to Bologna unifying system now a full‐time semester should correspond to 30 credit points.) Here, the first combination is accessible for the students with basic knowledge of calculus of single‐ and several‐variables, linear algebra and some Fourier analysis. The extended combination requires more of mathematical tools (in Chapter 2) and can be of interest for beginning graduates in applied math and engineering disciplines.
To conclude, the theory combined with approximation techniques and computer projects can give a better understanding of this useful tool (FEM) to solve differential equations. Finally, there are some easily implementable Matlab codes presented at the end of the book that are useful for freshmen in finite elements to test and check the theory through implementations.
Suggestions for possible course syllabus (what I have had)
| Chapters/Sections | 1 Semester | 7–8 wk | Credits |
| 1.1–1.4, 2.5, 2.6, 3, 5–7 | 3 h/wk | 6 h/wk | 5 |
| 1.1–1.4, 2.5, 2.6, 3, 5–10 | 4 h/wk | 8 h/wk | 7 |
| except 9.2, 9.3, 10.2.3–10.2.4, 10.5.4–10.5.6 | |||
| Whole material (includes Chapter 4) | 6 h/wk |
10 |
10 |
a) An intensive course for graduates in applied math/engineering.
All above configurations are associated with home and computer assignments. Examples of some assignments are given in Appendix C.
Acknowledgments
Many colleagues have been involved in the design, presentation, and correction of the material in this book. I wish to thank Niklas Eriksson and Bengt Svensson who have read the entire manuscript and made many valuable suggestions. Niklas has contributed to a better presentation of the text as well as to simplifications and corrections of many key estimates that has substantially improved the quality of the book. Bengt has made all xfig figures. The final version is further polished by John Bondestam Malmberg and Tobias Gebäck who, in particular, have many useful input in the Matlab codes. Many others have been involved in teaching and/or assisting me in teaching whole or parts of the content of the book as well as helping in design of computer assignments. They have supplied invaluable feedback and raised the quality of this work. Here are some (far from all): Fredrik Benzon, Christoffer Cromvik, Tommy Gustafsson, Kristin Kirchner, Fredrik Lindgren, Anders Logg, Fardin Saedpanah, Christoffer Standar, Maximilian Thaller. I owe all of them and the un‐named supporting colleagues my most sincere gratitude.
Mohammad Asadzadeh
Göteborg, Sweden
August 2019
1 Introduction
There are two ways of spreading light:
to be the candle or the mirror that reflects it.
Edith Wharton
This book presents an introduction to the Galerkin finite element method (FEM) as a powerful and general tool for approximating solution of differential equations. Our objective is twofold.
1 i) To present the main ordinary and partial differential equations (ODEs and PDEs) modeling different phenomena in science and engineering and introduce mathematical tools and environments for their analytic and numerical studies.
2 ii) To construct some common FEMs for approximate solutions of differential equations