A category theoretical interpretation of discretization in Galerkin finite element method

Valtteri Lahtinen*, Antti Stenvall

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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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.

Original languageEnglish
Pages (from-to)1271-1285
Number of pages15
JournalMATHEMATISCHE ZEITSCHRIFT
Volume296
Issue number3-4
Early online date1 Jan 2020
DOIs
Publication statusPublished - 1 Dec 2020
MoE publication typeA1 Journal article-refereed

Keywords

  • Category theory
  • Discretization
  • Engineering
  • Finite element method
  • Mathematical modeling

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