Why is there a difference between applying Laplacian operator vs. divergence to gradient of the data?

Laplacian operator will be equvalient to applying Divergence to the Gradient of the data. However, using the DEL2, DIVERGENCE, GRADIENT function in MATLAB, the results are different between the two methods. For example,

[x, y] = meshgrid(-1:.1:1);

A = sqrt(x.^2 + y.^2);

% the laplacian

Laplacian_1 = del2(A);

% second method to calculate the laplacian

[dx, dy] = gradient(A);

Laplacian_2 = divergence(dx, dy);

Laplacian_sq_error = (Laplacian_1 - Laplacian_2).^2;

The error is in the order of 1e-2.

Answer 1

[MathWorks Support Team]

MATLAB provides finite difference approximations to the partial differential operators gradient, divergence and Laplacian. With a spacing of size h in

     [x,y] = meshgrid(-1:h:1)

the finite difference approximations have accuracy proportional to h^2.