Mixed finite elements from the point of view of category theory

Abstract: I will first give an introduction to mixed finite elements by describing the main example (developed around 1980 by Raviart, Thomas and Nédélec but closely related to constructions in differential topology by de Rham, Weil and Whitney in the 1940ies). Then I will relate how many fundamental constructions in finite element theory can be described in terms of category theory. I will not assume prior knowledge of the latter. I will argue that the concept of a sheaf provides a unified framework for studying both continuous and discretized PDEs. Sheaf theory was first introduced by Leray in the 1940ies and later clarified in the Cartan seminar in the 1950ies with in particular algebraic geometry in mind. Applied sheaf theory is now a developing subject in many fields.

category theorydifferential geometrynumerical analysis

Audience: researchers in the topic

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CAM seminar

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