$3\mathrm{.}26$ Consider the steady$-$state heat transfer $($or other phenomenon$)$ in a square region shown

in Fig. P$3\mathrm{.}26.$ The governing equation is given by

$-\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(k\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}})-\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}(k\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}})=f_0$

The boundary conditions for the problem are:

$u(0,y)=y^2,u(x,0)=x^2,u(1,y)=1-y,u(x,1)=1-x$

Assuming $k=1$ and $f_0=2,$ determine the unknown nodal value of $u$ using the finite

element mesh shown in Fig. P$3\mathrm{.}26.$

Fig. P$3.26$