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2020 | OriginalPaper | Buchkapitel

Engineering Notes on Concepts of the Finite Element Method for Elliptic Problems

verfasst von : Jörg Schröder

Erschienen in: Novel Finite Element Technologies for Solids and Structures

Verlag: Springer International Publishing

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Abstract

In this contribution, we discuss some basic mechanical and mathematical features of the finite element technology for elliptic boundary value problems. Originating from an engineering perspective, we will introduce step by step of some basic mathematical concepts in order to set a basis for a deeper discussion of the rigorous mathematical approaches. In this context, we consider the boundedness of functions, the classification of the smoothness of functions, classical and mixed variational formulations as well as the \(H^{-1}\)-FEM in linear elasticity. Another focus is on the analysis of saddle point problems occurring in several mixed finite element formulations, especially on the solvability and stability of the associated discretized versions.

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Nächstes Kapitel Sensitivity Analysis Based Automation of Computational Problems

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nonnegative real values \(\mathbb {R}^+_0\), positive real values \(\mathbb {R}^+=\mathbb {R}^+_0\backslash 0\).

positive integers \(\mathrm{I\! N}_+=\{1,2,3,\dots \},\) nonnegative integers \(\mathrm{I\! N}_0=\{0,1,2,3,\dots \}=\mathrm{I\! N}_+ \cup \{0\}\).

The derivatives occurring in \(H^m({\mathcal B})\) have to be interpreted as weak or generalized derivatives. Classical derivatives are functions defined pointwise on an interval. A weak derivative need only to be locally integrable. If the function is sufficiently smooth, e.g., \(v\in C^m(\overline{{\mathcal B}})\), then its weak derivatives \(D^\alpha u\) coincide with the classical ones for \(|\alpha |\le m\).

Note that a restriction to homogeneous Dirichlet boundary conditions is only of technical nature and does not constitute a loss of generality, see, e.g., Braess (1997).

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Titel: Engineering Notes on Concepts of the Finite Element Method for Elliptic Problems
verfasst von: Jörg Schröder
Verlag: Springer International Publishing
Buch: Novel Finite Element Technologies for Solids and Structures
Print ISBN: 978-3-030-33519-9

Electronic ISBN: 978-3-030-33520-5

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-33520-5_1

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