High order approximation to non-smooth multivariate functions

Approximations of non-smooth multivariate functions return low-order approximations in the vicinities of the singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f = g + r+ where g, r ∈ C(R) and the function r+ is defined by r+(y) = { r(y), r(y) ≥ 0 0, r(y) < 0 , ∀y ∈ R . Given scattered (or uniform) data points X ⊂ R, we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.