How to find the best approximation for an optimality problem

Question

I have a problem in the following form: $$ J = \int_\Omega \left|\left|\frac{\partial f(W,x)}{\partial W}\cdot\gamma - N(f(W,x))\right|\right|^2dx $$ $N(f(W,x))$ is a nonlinear operator, function of $f$ and it's derivatives, that we suppose we can compute exactly and for every x. $W$ and $\gamma$ are vectors of the same lenght. Then, by the first order optimality condition: $$ \nabla_\gamma J=\int_\Omega \left(\frac{\partial f(W,x)}{\partial W}\cdot\gamma - N(f(W,x))\right)\frac{\partial f(W,x)}{\partial W} dx=\underline{0}. $$ I want to find which $\gamma$ provides the equality or, at least, minimizes a norm of the lhs. I can do this by discretizing the problem on a finite number of points. Now, let's call $\hat{\gamma}$ the solution obtained by solving the discretized problem, I would like to know if there is some clever way to choose a priori the position of a fixed number of discretization points $\{\bf{\bar{x}}\}$ in such a way that that: $$ \bf{\bar{x}}=\text{argmin}_{\bf{\bar{x}}}||\hat{\gamma} - \gamma|| $$ for some norm. Do you have any ideas?