Numerical reconstruction of Einstein's field equations

Question

A few analytic solutions are known to the Einstein field equations:

$$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R - kT_{\mu\nu} = 0$$

Taking a preexisting analytic solution such as Schwarzchild's solution:

$$g_{\mu\nu} = \begin{bmatrix} (1 - \frac{2m}{r}) & 0 & 0 & 0 \\ 0 & -\frac{1}{(1 - \frac{2m}{r})} & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -r^2\sin^2\theta \end{bmatrix}$$

Suppose we now delete a single arbitrary 2x2x2x2 grid locations' known $g_{\mu\nu}$ value. 

By what systematic method can we numerically recover the correct identical 10 numeric values of $g_{\mu\nu}$ in this grid, knowing only all of their neighboring points' analytic values except this 2x2x2x2 hole, forbidding any use of the analytic recipe that generates this numerical field?