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A category theoretical interpretation of discretization in Galerkin finite element method

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Details

Original languageEnglish
Number of pages15
JournalMATHEMATISCHE ZEITSCHRIFT
DOIs
Publication statusPublished - 2020
Publication typeA1 Journal article-refereed

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.

ASJC Scopus subject areas

Keywords

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

Publication forum classification

Field of science, Statistics Finland