Error norm for finite difference approximation I have approximated the differential equation using finite difference approximation and have the vector $$u$$. To find the error norm, it says I need the exact solution and the piecewise linear interpolant from $$u$$. Can anyone explain how I can get the piecewise linear interpolant? Would it be something like $$u_{i+1}-u_{i}$$ divided by I am not sure what. Also how do I get the exact solution? (Analytically??)

btw the bvp in question is:

$$-u'' + u' = 1$$ for $$0

$$u(0) = u(1) = 0$$

The basis solutions of the homogeneous equation are $$1, e^x$$, thus the particular solution is of the form $$ax$$ and by inserting $$a=1$$. The solution with the left boundary condition satisfied is thus $$u(x)=c(e^x-1)+x,$$ $$0=u(1)=c(e-1)+1$$ then gives $$c=-\frac1{e-1}$$.
The piecewise linear interpolant is a piecewise linear function that interpolates between the values of the numerical solution $$u_h((1-s)x_k+sx_{k+1})=(1-s)u_k+su_{k+1}~\text{ for all }~s\in[0,1].$$