# Discontinuous Galerkin FEM : Control points are mid-points of edges instead of nodes

I am thinking to use discontinuous galerkin FEM (DGFEM) method to estimate discontinuous displacement field $u: \Omega \rightarrow \mathbb{R}^2$ at the crack surface of a material.

The domain is discretized with triangles. I can think two ways to define control points where we evaluate displacement fields

1. Control points are defined on vertices/nodes of a triangle (probably knows as nodal DGFEM in the literature)
2. Control points are defined on mid-points of the edges. (like in Crouzeix–Raviart elements)

I could not find a reference where second method has been employed to model DGFEM.

Will there be something wrong mathematically if we define control points on mid-points of edges to model DGEM?

• With linear elements, if your points are at mid-point of edges, then the mass matrix is diagonal. I am not sure this is a big advantage since it is just a 3x3 matrix. – cfdlab Oct 28 '17 at 13:21
• Whether you choose vertices or mid-points, it is still called nodal DG since you are using Lagrange polynomials wrt those nodes. If instead you used some other basis set like Legendre polynomials, they are not associated to any nodes and are usually called modal DG scheme. – cfdlab Oct 28 '17 at 13:23
• @PraveenChandrashekar Thanks Praveen for the reply. – hari Oct 28 '17 at 14:36

## 1 Answer

For the discontinuous Galerkin method, the choice of control points is entirely unimportant because they conceptually lie in the interior of the cell (even if they are physically on the boundary of the cell). We think of a control point that is conceptually located at a vertex to be one that all cells adjacent to that vertex share; and of one on an edge that all cells adjacent to that edge share.

But in the DG methods, no control points are shared -- if you happen to physically locate a control point on an edge, then it will have to exist twice: one for each of the adjacent cells.

In other words, put your control points where it is convenient for you to define the function space on each cell.