nach oben

2024 | Buch

Kapitel lesen Erstes Kapitel lesen

Problems in Finite Element Methods

Aubin Nitsche’s Duality Process, Nodal Methods and Friedrichs Systems

verfasst von: Aref Jeribi

Verlag: Springer Nature Singapore

Buchreihe : Infosys Science Foundation Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Inhaltsverzeichnis

Über dieses Buch

This book discusses major topics and problems in finite element methods. It is targeted to graduate students and researchers in applied mathematics, physics, and engineering, wishing to learn and familiarize themselves with finite element theory. The book describes the nodal method for squares or rectangles and triangles, as well as an increase of the error between exact solution and approximate solution. It discusses an approximation of positive symmetric first-order systems in the Friedrichs sense by finite element methods. In addition, the book also explains the continuous and discontinuous approximation methods, adapted to the structure of the transport equation, leading to linear systems of quasi-explicit resolution, and therefore commonly used in practice.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The finite element method was born in the aeronautical industry, in the 1950s, more precisely around the research and development teams of Boeing. Only a few large companies in the sector have been able to acquire the digital means essential to the application of the method. The term “finite elements” is then widely developed. It has grown widely since the 1950s to pervade most areas of applied science, engineering, and financial markets. Nowadays, the finite element method is implemented in many academic or commercial software. It remains until now a blind simulation tool, therefore an object of study, and still knows many more specific developments (mobile or three-dimensional problems, applications on machines or massively parallel graphics cards, deformable meshes, etc.).

Aref Jeribi

Chapter 2. Fundamentals

Abstract

We begin by introducing here the mathematical notations and the basic results that will be used throughout this book. A (real) vector space is a set X, whose elements are called vectors, and in which two operations, addition and scalar multiplication, are defined as follows: (i) To every pair of vectors x and y corresponds a vector \(x + y\) in such a way that \(x + y = y + x\ \ \hbox {and} \ \ x + (y + z) = (x + y) + z.\) X contains a unique vector 0 (the zero vector or origin of X) such that \(x+0 = x\) for every \(x\in X\), and to each \(x\in X\) corresponds a unique vector \(-x\) such that \(x + (-x) = 0.\).

Aref Jeribi

Chapter 3. Variational Formulation of Boundary Problems

Abstract

In this chapter, we will focus on the notions of Sobolev spaces of order 1 and of order \(m\ge 1\). Sobolev spaces are functional spaces. More precisely, a Sobolev space is a vector space of functions provided with the norm obtained by the combination of the \(L^2\) norm of the function itself and of its derivatives up to a certain order. Derivatives are understood in a weak sense, within the meaning of distributions in order to make the space complete.

Aref Jeribi

Chapter 4. Introduction to Finite Element

Abstract

Once we have a weak formulation from a strong formulation, “you just have to” calculate the solution! The finite element method is one of the numerical tools developed to calculate an approximate solution (see [1]). The finite element method proposes to set up, on the basis of weak formulations, a discrete algorithm (discretization) making it possible to search for an approximate solution of a partial differential problem on a compact field with conditions at the edges and/or in inside the compact.

Aref Jeribi

Chapter 5. Non-conforming Methods

Abstract

Let \(\Omega \) be a polygonal domain of \(\mathbb {R}^2\) with piecewise smooth boundary \(\varGamma =\partial \Omega \). Given a function \(f\in L^2(\Omega ,\mathbb {R})\), we search a function u defined in \(\Omega \) checking

Aref Jeribi

Chapter 6. Nodal Methods

Abstract

Modern nodal methods were developed in the 1970s in the context of reactor calculations to solve the diffusion and transport problems of water reactors, both in stationary and transient states.

Aref Jeribi

Chapter 7. Positive Symmetric First-Order Systems Within the Meaning of Friedrichs

This chapter is devoted to the analysis of the positive symmetric first-order systems within the meaning of Friedrichs.

Aref Jeribi

Chapter 8. Approximation of the Transport Equation in Plane Two-Dimensional Geometry by Continuous and Discontinuous Finite Element Methods

Abstract

In this chapter, we study continuous and discontinuous approximation methods, adapted to the structure of the equations, leading to linear systems of quasi-explicit resolution and, therefore, commonly used in practice. All results of this chapter are due to P. Lesaint and may be found in [1].

Aref Jeribi

Chapter 9. Exercises with Solutions

Abstract

Demonstrate that if \((x_n)_n\) and \((y_n)_n\) are two sequences contained in the unit ball of a prehilbertian space such that.

Aref Jeribi

Chapter 10. Revision Issues

Abstract

Perform directly by finite difference techniques the increase of the error for the diagram.

Aref Jeribi

Backmatter

Titel: Problems in Finite Element Methods
verfasst von: Aref Jeribi
Verlag: Springer Nature Singapore
Electronic ISBN: 978-981-9757-10-7
Print ISBN: 978-981-9757-09-1
DOI: https://doi.org/10.1007/978-981-97-5710-7