``Exact'' numerical solution

The equation above form a linear system of equations which in principal could be solved for u_i,j for all points in the interior of the domain $\Omega$. This is a difficult set of equations to solve because of various reasons: first, there is no obvious ``pattern'' to the left and right hand sides, specially since the boundary values of the the field must be transferred to the known right hand side. Second, the number of unknown can be very large, of the order $N_x \times N_y$ in 2-D. A 3-D problem using a $1000 \times 1000 \times 1000$ lattice would involve 10⁹ equations!