# What are the degrees of freedom for BDM(1) elements in Fenics?

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Mixed Finite Element Methods and Applications by Boffi, Brezzi and Fortan describe the facet degrees of freedom as moments. For BDM(1) the degrees of freedom associated with certain edge are the zeroth and first moment of the finite element function on the edge. But Fenics does this in some different way and I have not been able to find how.

My motivation for this question comes from solving the mixed Poisson equation using BDM(1)-P(0) elements with Dirichlet boundary condition for BDM. From experimentation with boundary conditions it seems that the degrees of freedom are in fact function values in certain positions. Could you please confirm this and point me to relevant documentation or code, in the worst case? How will the degrees of freedom in Fenics look like for higher order BDM (and RT) elements?

My motivation for this question comes from solving the mixed Poisson equation using BDM(1)-P(0) elements with Dirichlet boundary condition for BDM. From experimentation with boundary conditions it seems that the degrees of freedom are in fact function values in certain positions. Could you please confirm this and point me to relevant documentation or code, in the worst case? How will the degrees of freedom in Fenics look like for higher order BDM (and RT) elements?

### 1 Answer

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In 2D, the degrees of freedom for BDM1 elements are the normal component of the vector field at two interior points (equidistantly spaced as far as I remember) for each edge of the reference triangle, weighted by the edge length.

See Chap 3 of the FEniCS book (Common and unusual finite elements), and in particular the illustrations for documentation. The illustrations also show the RT and higher order versions and ditto for 3D/tetrahedra. (Alternatively, you can look at the FIAT code for these elements.)
Thanks. It is a pity that there is no serious documentation for these topics.

See Chap 3 of the FEniCS book (Common and unusual finite elements), and in particular the illustrations for documentation. The illustrations also show the RT and higher order versions and ditto for 3D/tetrahedra. (Alternatively, you can look at the FIAT code for these elements.)

written
13 months ago by
TomL

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