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Consider the one-dimensional Poisson’s equation

$$−u''(x) + u(x) = f(x), \hspace{5mm} x \in (a, b),$$ with $u(a) = g_{1}$, $u'(b) = g_2$.

Discretize the equation using the finite element method with piecewise linear basis functions.

I am not sure what to do. For the discretization of f(x) I was taught to use the hat function but I am still not sure.

Any help would be greatly appreciated.

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Two references that should provide all necessary information, when combined together:

  1. Understanding Galerkin method of weighted residuals
  2. Are there any two-dimensional quadrature that only uses the values at the vertices of triangles?

Assuming that you can do yourself the details of the right hand side in $\,-u''(x)+u(x)=f(x)$ .

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