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# The Continuous Galerkin Finite Element Method Is Not Naturally Consistent with the Second Law of Thermodynamics

Version 1 : Received: 7 January 2020 / Approved: 9 January 2020 / Online: 9 January 2020 (07:18:53 CET)

How to cite: Limache, A.; Aimar, H. The Continuous Galerkin Finite Element Method Is Not Naturally Consistent with the Second Law of Thermodynamics. Preprints 2020, 2020010075 (doi: 10.20944/preprints202001.0075.v1). Limache, A.; Aimar, H. The Continuous Galerkin Finite Element Method Is Not Naturally Consistent with the Second Law of Thermodynamics. Preprints 2020, 2020010075 (doi: 10.20944/preprints202001.0075.v1).

## Abstract

It is well known that the Continuous Galerkin Finite Element (CGFE) method is globally consistent with respect to the first law of thermodynamics. This means that, for any mesh, all obtained discrete solutions will conserve total energy. One might expect, that the method is, also, globally consistent with respect to the second law of thermodynamics. In this paper, we formally study if such conjecture is true. The heat conduction equation is used as the physical model for this analysis. In the present study it is proved that the conjecture is false: at least, for standard piecewise linear (1D and 2D) elements, the CGFE method is not always globally consistent with respect to the second law of thermodynamics. In other words, some obtained discrete solutions can violate the global postulate of the second law, which asserts that total entropy can never decrease.

## Subject Areas

finite element method; second law of thermodynamics; heat equation; entropy

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