Integration of weak derivative of hat function for FEM.

Question

Just curious, lets say I want to solve

$u''=f$

$u(0)=0 \space\space u(1)=0$

In the weak form the equation looks like

$<u',\phi ' > = <f,\phi>$

Now my question, if I test my test functions $\phi_i$ to be hat functions. The weak derivatives will not be defined at grid points. When I do these inner products numerically, since there is a jump in the derivative of grid points, how am I suppose to use simpsons rule, quadrature, etc, when the function isn't defined at grid points.

Answer 1

The issues is I was using simpsons rule for the integration, which is troublesome if the function isn't well defined at the gridpoints like in this case. Using midpoint or gaussian quadature, which does not rely on the end points of the integral, does the trick.