Numerical differentiation for two-dimensional functions by a Fourier extension method

Based on the idea of Fourier extension, we develop a new method for numerical differentiation of two-dimensional functions on an arbitrary domain. The function will be extended to a periodic function on a larger domain. The Tikhonov regularization method in Hilbert scales is presented to deal with the ill-posedness of the problem. The Sobolev norm defined on the larger domain will be used as a stabilizer. An error analysis is provided with a regularization parameter chosen by a discrepancy principle. Numerical experiments are also presented to illustrate the effectiveness of the proposed method.