here's how you do it:
1) multiply your equation (the PDE) by the test function 2) integrate over the domain/cell 3) use integration by parts to separate volume and boundary terms 4) insert your expansion into the equation 5) that will give you a system to solve for your degrees of freedom u_h,i 6) evaluate the sum (the expansion) per element to get U_h 
Thank you for your response.
Indeed, this is the sixth point that interests me, i must evaluate the degree of freedom for use it in the sum. And for each item of the sum, i must raplace the function test by his value. It's that required ? 
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Hope that helps! 
In fact, what I meant is that after calculating the degrees of freedom we must return to the above formula of the sum to have the value of U in each element. I try to validate this point, because in some code I see U_h = U_h,n[n = max (i)] the last degrees of freedom

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To get U in your cell, just evaluate the (full) basis with the dof you computed....btw, are you referring to a modal or nodal scheme? 
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