Integral over a surface, given experimental data

I have a mesh of a 3D surface composed by triangles, and I have the value of a function $u(x,y,z)$ in every vertex of the mesh (every vertex of each triangle).

I need to calculate the following integral:

$iint_S u_{xx}dA$

Normally, I calculate this integral as a sum over each triangle supposing that $u$ is linear on each triangle (because I know the value in the vertex o each triangle). So I can use the basis polinomial (typical of fem). My problem is that this basis must be polynomials of degree 1, but I need the second partial derivate, so this integral will be zero.

How can I calculate this integral?