A category theoretical interpretation of discretization in Galerkin finite element method
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Mathematische Zeitschrift
A category theoretical interpretation of discretization in Galerkin finite element method Valtteri Lahtinen1,2 · Antti Stenvall2 Received: 26 September 2016 / Accepted: 7 January 2020 © The Author(s) 2020
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. Keywords Mathematical modeling · Category theory · Engineering · Finite element method · Discretization Mathematics Subject Classification 00A71 · 00A79 · 53Z05
1 Introduction Throughout engineering and physical sciences, field problems, arising from e.g. device design, are confronted. Apart from some exceptionally simple cases, such field problems cannot be solved analytically. Hence, in real modeling situations, they are solved numerically with a computer.1 One cannot, however, represent the solution in a function space that is 1 By analytical solution, we mean a solution which can be expressed in closed form. Numerical methods, on
the other hand, are approximation techniques that lead to a solution of a problem. However, even though it is often the case, this solution is not necessarily an approximation. A numerical solution can be exact, too. Valtteri Lahtinen is currently with Aalto University but this work was done during his post-doctoral period with Tampere University.
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Valtteri Lahtinen [email protected]
1
QCD Labs, QTF Centre of Excellence, Department of Applied Physics, Aalto University, 00076 Aalto, Finland
2
Electrical Engineering, Tampere University, PO Box 1001, 33014 Tampere, Finland
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V. Lahtinen, A. Stenvall
not finite-dimensional: Computers can only deal with a finite number of equations. Modelers, such as engineers and physicists, say that the problem has to be discretized. One of the most popular approximative numerical methods for discretizing and solving a field problem is the Galerkin finite element method (FEM) [3,16,18,19].2 In natural language, the word discrete refers to something separate and non-continuous.3 Hence, discretization is intuitively related to transferring from continuum to a state of separation, while maintaining enough information to utilize it in the sub-sequent decision making. In FEM, this is done by taking the function space in which the solution of the field problem is known to reside and then finding a suitable finite-dimensional subspace
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