CR Nonconforming P1 Element

We explain degree of freedoms for Crouzeix?Raviart nonconforming P1 element on triangles and tetrahedrons. The dofs are associated to edges (2-D) and faces (3-D). Given a mesh, the required data structure can be constructured by

[elem2edge,edge] = dofedge(elem);
[elem2face,face] = dof3face(elem);

Contents

Local indexing of DOFs

node = [1,0; 1,1; 0,0];
elem = [1 2 3];
edge = [2 3; 1 3; 1 2];
figure(1); clf;
set(gcf,'Units','normal');
set(gcf,'Position',[0,0,0.5,0.3]);
subplot(1,2,1)
showmesh(node,elem);
findnode(node);
findedge(node,edge);
node = [0,0,0; 1,0,0; 0,1,0; 0,0,1];
elem = [1 2 3 4];
face = [2 3 4; 1 3 4; 1 2 4; 1 2 3];
subplot(1,2,2)
showmesh3(node,elem);
view([-26 10]);
findnode3(node);
findelem(node,face);

The three dofs associated to edges in a triangle is displayed in the left and the four dofs associated to faces of a tetrahedron is in the right. The dofs are indexed such that the i-th dof is opposite to the i-th vertex.

Local bases of CR element

The d+1 Lagrange-type bases functions are denoted by $\phi_i, i=1:d+1$, i.e. $\phi_i(m_j)=\delta _{ij},i,j=1:d+1$. In barycentric coordinates, they are:

Global indexing of DOFs

node = [0,0; 1,0; 1,1; 0,1];
elem = [2,3,1; 4,1,3];
[node,elem] = uniformbisect(node,elem);
figure(2); clf;
showmesh(node,elem);
findnode(node);
findelem(node,elem);
[elem2edge,edge] = dofedge(elem);
findedge(node,edge);

The matrix elem2edge is the local to global index mapping of edges.

display(elem2edge);

elem2edge =

           5           6          15
          10          11          14
           2           1          13
           7           9          16
           7          15           8
           2          14           3
           5          13           4
          10          16          12


Published with MATLAB® R2014b