Finite element assembly

Finite element assembly adds local element matrices and load vectors into a global linear system according to the correspondence between local and global degrees of freedom. A shared node receives contributions from every incident finite element. Boundary integrals, such as those in a Robin boundary condition, are assembled from local boundary-edge matrices in the same way.

Triangulate the region $0\le x,y\le 2$, $x+y\ge 2$ by $x=1$, $y=1$, and $x+y=3$. Impose zero Dirichlet data on the horizontal and vertical exterior edges and zero Neumann data on the sloping edge. Only the midpoint $B=(1,1)$ of the sloping edge remains free in the continuous piecewise linear space. For source one, its stiffness entry is $1/2+1/2+1=2$, while its load is three triangle contributions of $1/6$. The single free value is therefore $1/4$.