A category theoretical interpretation of discretization in Galerkin finite element method
Lahtinen, Valtteri; Stenvall, Antti (2020)
Lahtinen, Valtteri
Stenvall, Antti
2020
MATHEMATISCHE ZEITSCHRIFT
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:tuni-202005285768
https://urn.fi/URN:NBN:fi:tuni-202005285768
Kuvaus
Peer reviewed
Tiivistelmä
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.
Kokoelmat
- TUNICRIS-julkaisut [16983]