Title: 2. Introduction to the Finite Element Method
Abstract: Previous chapter Next chapter Classics in Applied Mathematics The Finite Element Method for Elliptic Problems2. Introduction to the Finite Element Methodpp.36 - 109Chapter DOI:https://doi.org/10.1137/1.9780898719208.ch2PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt Introduction The basic scope of this chapter is to introduce the finite element method and to give a thorough description of the use of this method for approximating the solutions of second-order or fourth-order problems posed in variational form over a space V. A well-known approach for approximating such problems is Galerkin's method, which consists in defining similar problems, called discrete problems, over finite-dimensional subspaces Vh of the space V. Then the finite element method in its simplest form is a Galerkin's method characterized by three basic aspects in the construction of the space Vh: First, a triangulation Jh is established over the set Ω ¯ , i.e., the set Ω ¯ is written as a finite union of finite elements K ∈ Jh . Secondly, the function υh ∈ Vh are piecewise polynomials, in the sense that for each K ∈ Jh , , the spaces PK = {υh∣K; υh ∈ Vh} consist of polynomials. Thirdly, there should exist a basis in the space Vh whose functions have small supports. These three basic aspects are discussed in Section 2.1, where we also give simple criteria which insure the validity of inclusions such as Vh ⊂ H1(Ω), Vh ⊂ H01(Ω), etc… (Theorems 2.1.1 and 2.1.2). We also briefly indicate how the three basic aspects are still present in the more general finite element methods to be subsequently described. In this respect, we shall reserve the terminology conforming finite element method for the simplest such method (as described in this chapter). Previous chapter Next chapter RelatedDetails Published:2002ISBN:978-0-89871-514-9eISBN:978-0-89871-920-8 https://doi.org/10.1137/1.9780898719208Book Series Name:Classics in Applied MathematicsBook Code:CL40Book Pages:xxiii + 529Key words:differential equations, elliptics, numerial solutions, boundary value problems, finite element method
Publication Year: 2002
Publication Date: 2002-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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Cited By Count: 1
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