An efficient interpolation algorithm on anisotropic grids for functions with jump discontinuities in 2-D

In this paper we construct an algorithm that generates a sequence of continuous functions that approximate a given real valued function f of two variables that have jump discontinuities along a closed curve. The algorithm generates a sequence of triangulations of the domain of f. The triangulations include triangles with high aspect ratio along the curve where f has jumps. The sequence of functions generated by the algorithm are obtained by interpolating f on the triangulations using continuous piecewise polynomial functions. The approximation error of this algorithm is O(1/N2) when the triangulation contains N triangles and when the error is measured in the L1 norm. Algorithms that adaptively generate triangulations by local regular refinement produce approximation errors of size O(1/N), even if higher-order polynomial interpolation is used.