P1 Linear Element

For the linear element on a simplex, the local basis functions are barycentric coordinate of vertices. The local to global pointer is elem. This is the default element for elliptic equations.

Contents

A local basis of P1

For $i = 1, 2,..., d+1$, the local basis of linear element space is

$$\phi_i = \lambda_i, \nabla \phi_i = \nabla \lambda_i = - \frac{|e_i|}{d!|T|}\mathbf n_i,$$

where $e_i$ is the edge opposite to the i-th vertex and $n_i$ is the unit outwards normal direction.

See Finite Element Methods Section 2.1 for geometric explanation of the barycentric coordinate.

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;
showmesh(node,elem);
findnode(node);
findelem(node,elem);
display(elem);

elem =

     8     6     2
     7     6     4
     5     6     1
     9     6     3
     8     3     6
     7     1     6
     5     2     6
     9     4     6


Published with MATLAB® R2014b