Hello Mathematics Stack Exchange Community,
I'm working on a numerical problem where I have the first-order partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of a bivariate function $f(x, y)$ at a set of points $(x_i, y_i)$. My goal is to approximate the mixed second-order partial derivative $\frac{\partial^2 f}{\partial x \partial y}$ at these points using the Radial Basis Function-generated Finite Differences (RBF-FD) method.
As I understand it, I could compute this in two ways. I could either apply RBF-FD to the given $\frac{\partial f}{\partial x}$ values with respect to $y$, or I could apply it to the $\frac{\partial f}{\partial y}$ values with respect to $x$. Mathematically, I expect the mixed derivatives $\frac{\partial^2 f}{\partial x \partial y}$ and $\frac{\partial^2 f}{\partial y \partial x}$ to be equal, as per the symmetry of the Hessian for sufficiently smooth functions. However, these two methods might not necessarily agree in their results.
I am seeking advice on how to approach this problem to ensure that the approximated mixed derivatives respect this symmetry. Is there a recommended method to modify or combine these approaches in the RBF-FD framework to achieve consistent results?
Any insights, references, or suggestions on how to tackle this issue would be greatly appreciated.
Thank you!
