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Finite Element Method

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Numerical Approximation of Partial Differential Equations

Part of the book series: Texts in Applied Mathematics ((TAM,volume 64))

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Abstract

Finite element methods provide an abstract framework for interpolating functions or vector fields in multidimensional domains. They allow for specifying Galerkin methods for approximating partial differential equations. In combination with regularity results, error estimates in various norms can be proved. The efficient implementation of low order and isoparametric methods is discussed in the case of stationary and evolutionary model problems.

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Notes

  1. 1.

    A discussion of the historical development of the finite element method can be found in [9]. The monograph [5] provides a comprehensive overview over finite element methods and their analysis. Constructive existence theories for interpolating finite element functions are stated in [3]; a more practical approach is followed in [2]. An important contribution to the numerical analysis of finite element methods for parabolic equations is the article [22]; further results can be found in the monograph [21]. Classical articles on error estimates for finite element methods are the references [1, 14, 15, 17, 18]; see [6] for quadrature rules on simplices. Further references on finite element methods for partial differential equations are [4, 7, 8, 10–13, 16, 19, 20].

References

A discussion of the historical development of the finite element method can be found in [9]. The monograph [5] provides a comprehensive overview over finite element methods and their analysis. Constructive existence theories for interpolating finite element functions are stated in [3]; a more practical approach is followed in [2]. An important contribution to the numerical analysis of finite element methods for parabolic equations is the article [22]; further results can be found in the monograph [21]. Classical articles on error estimates for finite element methods are the references [1, 14, 15, 17, 18]; see [6] for quadrature rules on simplices. Further references on finite element methods for partial differential equations are [4, 7, 8, 10–13, 16, 19, 20].

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  1. Angewandte Mathematik, Albert-Ludwigs-Universitaet, Freiburg, Germany

    Sören Bartels

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Bartels, S. (2016). Finite Element Method. In: Numerical Approximation of Partial Differential Equations. Texts in Applied Mathematics, vol 64. Springer, Cham. https://doi.org/10.1007/978-3-319-32354-1_3

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