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

  • Jörg SchröderEmail author
Chapter
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 597)

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.

Notes

Acknowledgements

We thank the DFG for the financial support within the SPP 1748 Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis, project Novel finite elements— Mixed, Hybrid and Virtual Element formulations (Projectnumber: 255432295) (SCHR 570/23-2). I would also like to thank Nils Viebahn for helpful discussions and his help with the manuscript and Sascha Maassen and Rainer Niekamp for the implementation of the \(H^{-1}\)procedure and accompanying discussions.

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Copyright information

© CISM International Centre for Mechanical Sciences 2020

Authors and Affiliations

  1. 1.Department of Civil Engineering, Faculty of Engineering, Institute of MechanicsUniversity of Duisburg-EssenEssenGermany

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