A Simple Introduction to the Mixed Finite Element Method

Theory and Applications

verfasst von: Gabriel N. Gatica

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Mathematics

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

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Über dieses Buch

The main purpose of this book is to provide a simple and accessible introduction to the mixed finite element method as a fundamental tool to numerically solve a wide class of boundary value problems arising in physics and engineering sciences. The book is based on material that was taught in corresponding undergraduate and graduate courses at the Universidad de Concepcion, Concepcion, Chile, during the last 7 years. As compared with several other classical books in the subject, the main features of the present one have to do, on one hand, with an attempt of presenting and explaining most of the details in the proofs and in the different applications. In particular several results and aspects of the corresponding analysis that are usually available only in papers or proceedings are included here.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In this chapter we base most of the presentation on the classical references [8, 20, 41, 51] and describe the main introductory aspects of the finite and mixed finite element methods.

Gabriel N. Gatica

Chapter 2. Babuška–Brezzi Theory

Abstract

In this chapter we present the main results forming part of the Babuška–Brezzi theory, which makes it possible to analyze a large family of mixed variational formulations and their respective Galerkin approximations. Our main references here include [16, 41, 50, 52]. We begin by introducing the specific kind of operator equations that we are interested in.

Gabriel N. Gatica

Chapter 3. Raviart-Thomas Spaces

Abstract

In this chapter we introduce Raviart–Thomas spaces, which constitute the most classical finite element subspaces of \(H(\mathrm{div};\varOmega )\), and prove their main interpolation and approximation properties. Several aspects of our analysis follow the approaches from [16, 50, 52].

Gabriel N. Gatica

Chapter 4. Mixed Finite Element Methods

Abstract

In this chapter we utilize the Raviart–Thomas spaces to present and analyze specific mixed finite element methods applied to some of the examples studied in Chap. 2. The corresponding discussion follows mainly the presentations in [12, 39, 50, 52]. We begin with a preliminary section dealing with the approximation properties of the finite element subspaces to be employed.

Gabriel N. Gatica

Backmatter

Titel: A Simple Introduction to the Mixed Finite Element Method
verfasst von: Gabriel N. Gatica
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-03695-3
Print ISBN: 978-3-319-03694-6
DOI: https://doi.org/10.1007/978-3-319-03695-3