# FEM: Testing basis functions in a subspace V_h

In the finite element method, at a certain point we arrive at the following Galerkian problem where it is desired to find the solution $$u_h \space \in V_h$$ that solves the following equation:

$$a(u_h,v_h)=L(v_h) \space \space \space \forall v_h \in V_h$$

where $$a$$ and $$L$$ are, respectively, a bilinear and linear operators. I cannot understand why is normally stated that it is enough to test against a set of basis functions $$\Phi_i \in V_h$$ (which are linearly combined to form $$u_h$$)and not against all functions $$v\in V_h$$

Thank you very much in advance and I hope you may help me understanding this issue.

Kind regards

The idea of FEM is to find a finite set of equations after getting the Galkerin form. Once we have reached $$a(u_h,v_h)=L(v_h),$$ we want to reduce it to a system of equations of the form $$Au=L$$. Now an easy way to do this is if we choose $$v_h=\{\phi_i\}$$ because we have finite dimension and hence finite $$\{\phi_i\}$$, and the biggest thing is that we know these elements.
Now, the question is why only $$\{\phi_i\}$$ and not any other $$n$$ elements of $$V_h$$. Well if we choose $$\phi_i$$ then we can easily compute our matrix $$A=\{a_{ij}\}$$ as then we have $$a_{ij}=a(\phi_i,\phi_j)$$ and we already know these elements.