I am still new to finite element methods, and I was looking at some tutorials on a specific formulation of the Poisson equation that introduces an additional variable. Some of the tutorials call this the mixed poisson formulation, though I am not sure if that is a standard term. Here is the equation.
$$ \sigma - \nabla u = 0 $$ $$ \nabla \cdot \sigma = -f $$
I was looking at tutorials for solving this system in both Fenics, and Deal II, which are widely used FEM packages. Both of the tutorials talk about needed Brezzi-Douglas-Marini elements or Hdiv elements for solving this system numerically.
So my question is two fold.
What is the difference between a standard Lagrange element and a BDM or Hdiv element? I am just not clear on what the mathematical difference is between these elements. I was looking at this article on the topic, but the notation was a little dense for me to read without some existing familiarity with the topic.
Why would someone use a BDM or Hdiv element to solve this problem, versus using a Lagrange element in 2 or more dimensions--depending on the problem?
Please note, I did consider asking this question on the Computational Science SE, but I checked and there are no existing questions on that site with reference to Hdiv or BDM elements. Hence, I thought that a wider numerical methods audience might be a more appropriate audience for this question.
Thanks.
First to answer your second question. People use finite elements specialized to the $H(\mathrm{div})$ space because they 'work' while the Lagrange elements do not 'work' for the problem you described. Lagrange elements typically introduce some kind of pathology to the discretized problem that is absent while using specialized $H(\mathrm{div})$ elements like $BDM$. This is usually caused by violating the so called discrete $\inf\sup$ condition.
The $\inf\sup$ condition is a common tool to prove that the saddle point problem you mention has a solution on both functional and discrete level. If the condition does not hold for your discretization or even depends on mesh parameter you can not expect to get meaningful results from the solution of the discretized system. $BDM$ elements are designed to satisfy the discrete $\inf\sup$ condition so they are safe to use. The saddle point problem has a structure (certain properties of the divergence operator in the case of your example problem) that the discretization has to respect.
The difference in Lagrange and $BDM$ elements is given by their construction. They have different degrees of freedom and related basis functions and different properties. $BDM$ elements have continuous normal component over edges of a mesh while Lagrange elements are continuous over edges. You can check out the basis functions for Lagrange elements and $BDM$ on defelement.com.
Most of the details to this are in the scriptum you linked, another relatively straigthforward introduction can be found in A Simple Introduction to the Mixed Finite Element Method.