| Kommentar |
Many physical phenomena, such as heat conduction, electrostatic fields or elastic deformations of solid bodies, can be described through elliptic partial differential equations. Only in special cases can the solutions of these equations be obtained using analytic methods. Hence, provably efficient and reliable numerical methods are required to compute approximate solutions. In this course, we will be concerned with the numerical analysis of so-called finite-element methods, which have an elegant mathematical theory and are wildly adopted in academic and industrial applications. Our focus will lie on establishing convergence rates for these methods and deriving rigorous and efficiently computable a posteriori error bounds. Further, we will consider extensions to time-dependent problems and equation systems.
In the accompanying exercises, in addition to deepening our theoretical knowledge, we will gain practical experience with the implementation of finite element methods using the Python programming language. |
| Literatur |
- Hans Wilhelm Alt. Lineare Funktionalanalysis. Springer Berlin Heidelberg, 2012.
- Hans Wilhelm Alt. Linear Functional Analysis. Universitext. Springer London, 2016.
- Dietrich Braess. Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer Berlin Heidelberg, 2007.
- Dietrich Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, 2007.
- Susanne C. Brenner and L. Ridgway Scott. The mathematical theory of finite element methods. Vol. 15. Texts in Applied Mathematics. New York: Springer, 2008.
- Philippe G. Ciarlet. The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, 2002.
- Alexandre Ern and Jean-Luc Guermond. Theory and Practice of Finite Elements. Springer New York, 2004.
- Alexandre Ern and Jean-Luc Guermond. Finite Elements I: Approximation and Interpolation. Vol. 72. Texts in Applied Mathematics. Springer International Publishing, 2021.
- Alexandre Ern and Jean-Luc Guermond. Finite Elements II: Galerkin Approximation, Elliptic and Mixed PDEs. Vol. 73. Texts in Applied Mathematics. Springer International Publishing, 2021.
- Alexandre Ern and Jean-Luc Guermond. Finite Elements III: First-Order and Time-Dependent PDEs. Vol. 74. Texts in Applied Mathematics. Springer International Publishing, 2021.
- Alexandre Ern and Jean-Luc Guermond. Finite Elements IV: Exercises and Solutions. 2021. url: https://hal.archives-ouvertes.fr/hal-03226052.
- Yousef Saad. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, 2003.
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