Piecewise Padé-Chebyshev Approximation of Bivariate Piecewise smooth Functions

This article aims to implement the novel piecewise Maehly based Pad\'e-Chebyshev approximation and study its utility in minimizing the Gibbs phenomenon while approximating piecewise smooth functions in two-dimensions. We first develop a piecewise Pad\'e-Chebyshev method (PiPC) to approximate univariate piecewise smooth functions and then extend the same to a two dimensional space, leading to a piecewise bivariate Pad\'e-Chebyshev approximation (Pi2DPC) for approximating bivariate piecewise smooth functions. The chief advantage of these methods lie in their non dependence on any apriori knowledge of the locations and types of singularities present in the original function. Finally, we supplement our method with numerical results which validate its effectiveness in diminishing the Gibbs phenomenon to negligible levels.