Abstract
This chapter presents, first, an overview of the mathematical tools required to undertake studies of the well-posedness of linear boundary value problems and their approximations by finite elements. In the remainder of this work, these tools are used to examine the existence and uniqueness of solutions to weak boundary value problems, and convergence of finite element approximations. The emphasis is on second-order partial differential equations, with the governing equations for linear elasticity being the key model problem.
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Notes
- 1.
A connected set \(\Omega \) is one for which every pair of points can be connected by a curve that lies entirely in \(\Omega \); \(\Omega \) is open if it comprises only interior points; and a domain is an open connected set.
- 2.
Strictly speaking, we should define these as weak derivatives: see Reddy (1998).
References
Atkinson, K., & Han, W. (2001). Theoretical numerical analysis: A functional analysis framework. Heidelberg: Springer.
Ciarlet, P. G. (2002). The finite element method for elliptic problems. Classics in applied mathematics. Philadelphia: Society for Industrial and Applied Mathematics.
Reddy, B. D. (1998). Introductory functional analysis: With applications to boundary value problems and finite elements. Heidelberg: Springer.
Acknowledgments
The support of the South African Department of Science and Technology and National Research Foundation through the South African Research Chair in Computational Mechanics is gratefully acknowledged.
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© 2016 CISM International Centre for Mechanical Sciences
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Reddy, B.D. (2016). Functional Analysis, Boundary Value Problems and Finite Elements. In: Schröder, J., Wriggers, P. (eds) Advanced Finite Element Technologies. CISM International Centre for Mechanical Sciences, vol 566. Springer, Cham. https://doi.org/10.1007/978-3-319-31925-4_1
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DOI: https://doi.org/10.1007/978-3-319-31925-4_1
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