Dgdrxrt

What is the discrete (finite difference) Laplacian operator $Delta_h$ in $mathbbR^N$? What is the corresponding matrix $A_h$ such that
$Delta_h f = A_h cdot f$?

On Wikipedia and on most textbooks only one or two-dimensional case is considered:
$$Delta_h f(x,y)approx frac f(x-h,y)+f(x+h,y)+f(x,y-h)+f(x,y+h)-4f(x,y)h^2.$$

In one dimension we have the approximation for the second derivative:

$$f^(2)(x)approx fracf(x+h) + f(x-h) + 2h^2$$

This then gives you the approximation for the second partial derivative in some particular direction, the Laplacian is the sum of the second derivatives w.r.t. all the coordinates.

There are many different possible finite difference approximations of many different orders of accuracy. In numerical methods a common choice is to use ''centered in space:''
$$f(x+h) = f(x) + f'(x)h + f''(x)frach^22 + f'''(x)frach^36 + f''''(x)frach^424$$

$$f(x-h) = f(x) - f'(x)h + f''(x)frach^22- f'''(x)frach^36+f''''(x)frach^424$$

$$Longrightarrowfracf(x+h) + f(x-h) - 2f(x)h^2= f''(x) + f''''(xi)frach^212 $$
$$Longrightarrow f''(x) = fracf(x+h) + f(x-h) - 2f(x)h^2 + mathcalO(h^2)$$

This will correspond to a differentiation matrix:
$$D_+-xf(x) =
frac1h^2beginpmatrix
&-2 & 1& &\
&1 &ddots &ddots & \
& &ddots & ddots & 1 \
& & & 1 & -2
endpmatrixbeginpmatrixf_1(x) \ f_2(x) \ vdots \ f_n-1(x) \ f_k(x) endpmatrix$$
where $k$ is total number of grid points used

what you are looking at is the analogous idea for a 2 variable function

In general, the formula for $f(x_1,dots,x_n)$ will be:

$$nabla f = sum_j=1^n fracpartial^2 fdx_j^2 = left(sum_j=1^n D_+-x_jright)f +mathcalO(h^2)$$ (assuming you use the same grid spacing for each variable)