Mixed formulation in 1D

Question

I have been working on a hybrid dimensional model using the mixed FEM formulation, in which 3D elements and 2D elements are combined by certain relationships between the degrees of freedom (DOFs) between face flux for 3D elements and pressure DOFs for 2D elements. I'm doing this for Raviart Thomas elements of various orders, i.e., RT0-RT1 etc. However, I'm struggling to find the 1D equivalent of the RT elements. I have not found any reference about 1D mixed formulation elements, let alone 1D RT elements.

I understand that the lowest 1D mixed element would be quite simple: 1 DOF for the constant pressure and 2 DOFs for the velocity field, one for each end node of the element. But what about higher order elements? Pressure DOFs will surely be point evaluations along the interval, but how can the DOFs for the velocity be defined?

Any help or hint will be greatly appreciated.

Answer 1

The RT elements for a given dimension span $H(div)$, the space of vector fields for which the components and the weak divergence are square-integrable. In one dimension the distinctions between vectors and scalars break down, and this is just the space of square-integrable functions with square integrable weak derivatives, otherwise known as $H^1$. As such, all your 1D elements are likely to be common Lagrange elements of various polynomial degrees, with or without continuity across element boundaries.

From what you describe, it's possible you may be interested in the $P2-P1$ discontinuous element pair, where velocities are piecewise quadratic and continuous (for the nodal representation, think a DOF at each end of the interval, and one in the middle) and pressure is piecewise linear, but discontinuous (you can put the DOFs where you like for that one). This is sometimes described in 2D for the incompressible Stokes problem, but a 1D description isn't very interesting, since the only incompressible 1D velocity fields are constant everywhere.