Discrete and conforming smooth de Rham complexes in three dimensions

Neilan, Michael

doi:10.1090/S0025-5718-2015-02958-5

Discrete and conforming smooth de Rham complexes in three dimensions
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Math. Comp. 84 (2015), 2059-2081 Request permission

Abstract:

Conforming discrete de Rham complexes consisting of finite element spaces with extra smoothness are constructed. In particular, we develop $H^2$, $\boldsymbol {H}^1({\text {curl}})$, $\boldsymbol {H}^1$ and $L^2$ conforming finite element spaces and show that an exactness property is satisfied. These results naturally lead to discretizations for Stokes and Brinkman type problems as well as conforming approximations to fourth order curl problems. In addition, we reduce the question of stability of the three-dimensional Scott-Vogelius finite element to a simply stated conjecture.

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