This is accomplished by approximating u in (1) in terms of a finite basis, a spline basis. To derive the formula for the FEM, we first multiply the equation by an element of dom(E), and integrate:

Notice that -т_SG² Du v dm = E(u,v) by the weak definition of the Laplacian. We approximate u by its projection into the spline space, and can represent it as a sum in terms of the spline basis: u = е_{f О S1} c_f f. Further, we can also require v to be in S₁ instead of just dom(E), so our equations become:

where each f_i is a basis element of S₁.