A category theoretical interpretation of discretization in Galerkin finite element method

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2020-12-01
Department
Department of Applied Physics
Tampere University
Major/Subject
Mcode
Degree programme
Language
en
Pages
15
1271-1285
Series
MATHEMATISCHE ZEITSCHRIFT, Volume 296, issue 3-4
Abstract
The Galerkin finite element method (FEM) is used widely in finding approximative solutions to field problems in engineering and natural sciences. When utilizing FEM, the field problem is said to be discretized. In this paper, we interpret discretization in FEM through category theory, unifying the concept of discreteness in FEM with that of discreteness in other fields of mathematics, such as topology. This reveals structural properties encoded in this concept: we propose that discretization is a dagger mono with a discrete domain in the category of Hilbert spaces made concrete over the category of vector spaces. Moreover, we discuss parallel decomposability of discretization, and through examples, connect it to different FEM formulations and choices of basis functions.
Description
Keywords
Category theory, Discretization, Engineering, Finite element method, Mathematical modeling
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Citation
Lahtinen , V & Stenvall , A 2020 , ' A category theoretical interpretation of discretization in Galerkin finite element method ' , MATHEMATISCHE ZEITSCHRIFT , vol. 296 , no. 3-4 , pp. 1271-1285 . https://doi.org/10.1007/s00209-020-02456-1