Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / B / 6 / iii / Solution

Write $u_h=U_B{\varphi }_B$. On $ABD,BCE,BDE$, the gradients are $(0,-1)$, $(-1,0)$ and $(-1,-1)$, respectively. ThusThe integral of an affine nodal function over a triangle equals its area times the mean of its three vertex values. Each supported triangle therefore contributes $1/6$ to $\int {\varphi }_B$, giving total $1/2$. The reduced system is $[2][U_B]=[1/2]$, and the complete nodal answer isOnly one algebraic unknown remains despite the two-dimensional domain.