Numerical Differentiation of 2D Functions by a Mollification Method Based on Legendre Expansion

In this paper, we consider numerical differentiation of bivariate functions when a set of noisy data is given. A mollification method based on spanned by Legendre polynomials is proposed and the mollification parameter is chosen by a discrepancy principle. The theoretical analyses show that the smoother the genuine solution, the higher the convergence rates of the numerical solution. To get a practical approach, we also derive corresponding results for Legendre-Gauss-Lobatto interpolation. Numerical examples are also given to show the efficiency of the method.