# [deal.II] Question about the numbering of DoFs

```Dear all,
I am reading the page :
https://www.dealii.org/current/doxygen/deal.II/classFiniteElement.html.
I feel confused by the numbering of the degrees of freedom. For example:```
```
FESystem<dim>
<https://www.dealii.org/current/doxygen/deal.II/classFESystem.html>
fe_basis(FE_Q<dim>
<https://www.dealii.org/current/doxygen/deal.II/classFE__Q.html>(2), dim,
FE_Q<dim> <https://www.dealii.org/current/doxygen/deal.II/classFE__Q.html>
(1),1);

As mentioned in Glossary:

Describing "degrees of freedom" in this context requires us to simply
*enumerate* the basis functions of the space *V**h*. For *Q*1 elements this
means simply enumerating the vertices of the mesh in some way, but for
higher elements one also has to enumerate the shape functions that are
associated with edges, faces, or cell interiors of the mesh.

We have a vector u which has 2 components defined on 9 nodes. On each node,
there are 2 base functions, so the degrees of freedom are 9*2*2 = 36. And
we have a scalar p which has 1 component defined on 4 nodes. On each node,
there is one base function, so the degrees of freedom are 4*1*1=4. The
total degrees of freedom should be 36+4=40.

In the webpage, it gives 2*9+4*1=22 degrees of freedom, it equals the base
function with the vector components.
[image: image.png]

u(0) should have two base element [phi(0),0] and [phi(0) 0], u(1) should
have two base elements as well [0 phi(0)] [0 phi(1)], so here should be 4
rather than 2 degrees of freedom.

Best regards,

--
Yuesu Jin,
Ph.D student,
University of Houston,
College of Natural Sciences and Mathematics,
Department of Earth and Atmospheric Sciences,
Houston, Texas 77204-5008
346-404-2062

--
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