Meshless Methods and the Kronecker Delta Property

In texts I often read that meshless methods, as opposed to the FEM, do not exhibit the Kronecker Delta Property [1]:

$N_I(x_J)\neq\delta_{IJ} = \begin{cases}1 &\mbox{if } I=J \\0 &\mbox{otherwise}\end{cases}$

Where $N_I$ is the meshless shape function at position $x_I$. I understand this property, however, I'm not sure of the implications. Say, one wants to solve a problem in elastostatics of the form

$\mathbf{Kf}=\mathbf{u}$

Where $\mathbf{K}$ is the stiffness Matrix, $\mathbf{f}$ the force vector and $\mathbf{u}$ the displacement. The construction of those matrices would be analogous to FEM, except the for using the meshless shape functions, see e.g. [2].

Now, also in [2] (and in in other texts) it is claimed that, when solving for $\mathbf{u}$, the solution is in fact a fictitious one and the real solution needs to be found by re-approximation using the shape functions:

$u_I = \sum\limits_JN(x_J)u_J$

However, I don't understand how the use of shape functions that do not satisfy the Kronecker delta property entails the need for this step. Can anybody provide me some insight?

In the general case, your system of equations is posed for the coefficients of the expansion $$u(x) \approx \sum_{i=1}^N c_i f_i(x)$$ i.e., for $c_i$.
Let us expose this with an example using the Ritz method. We are trying to solve $$\frac{d^2 u}{dx^2} + u + x = 0$$ with boundary conditions $u(0)=u(1)=0$. Let us use the approximate function $$u_2(x) = x(1-x)[c_1 + c_2 x]$$ The functional for this approximation is $$J[u_2] = \frac{13}{210}c_2^2 + \frac{3}{20}c_1 c_2 - \frac{1}{2}c_2 + \frac{3}{20}c_1^2 - \frac{1}{12}c_1 \enspace .$$ Imposing the stationarity ($\delta J=0$) we obtain the system of equations \begin{align} &126 c_1 + 63 c_2 = 35\\ &63 c_1 + 52 c_2 = 21 \end{align} that has as solution $c_1 = 71/369$ and $c_2=7/41$. Notice that these two coefficients are just numbers, and they do not represent a function over our domain $[0,1]$. They play the role of multiply the functions of our expansion, that give us as solution $$u \approx u_2(x) = x(1-x)\sum_{i=1}^2 c_i x^{i-1} = (1 - x)x\left(\frac{7}{41}x + \frac{71}{369}\right) \enspace .$$