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Finite sections: A functional analytic perspective on approximation methods
Author(s) -
Lindner Marko,
Seifert Christian
Publication year - 2018
Publication title -
gamm‐mitteilungen
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.239
H-Index - 18
eISSN - 1522-2608
pISSN - 0936-7195
DOI - 10.1002/gamm.201800013
Subject(s) - finite element method , focus (optics) , stability (learning theory) , separable space , mathematics , section (typography) , hilbert space , perspective (graphical) , galerkin method , polynomial , mixed finite element method , mathematical analysis , calculus (dental) , computer science , physics , geometry , medicine , dentistry , machine learning , optics , thermodynamics , operating system
We review approximation methods and their stability and applicability. We then focus on the finite section method and Galerkin methods and show that on separable Hilbert spaces either one can be interpreted as the other. In the end we demonstrate that well‐known methods such as the finite element method and polynomial chaos expansion are particular examples of the finite section method; their applicability can therefore be studied via the latter.