Interpolation and approximation of piecewise smooth functions with corner discontinuities on sigma quasi-uniform grids
This work addresses the challenge of reducing approximation errors for functions with isolated singularities, which is incremental but useful for applications like shock capturing and image processing.
The paper tackles the problem of approximating piecewise smooth functions with corner discontinuities using nonlinear interpolation on sigma quasi-uniform grids, achieving optimal approximation order for grids with spacing below a critical value h_c.
This paper provides approximation orders for a class of nonlinear interpolation procedures for univariate data sampled over $Ï$ quasi-uniform grids. The considered interpolation is built using both essentially nonoscillatory (ENO) and subcell resolution (SR) reconstruction techniques. The main target of these nonlinear techniques is to reduce the approximation error for functions with isolated corner singularities and in turn this fact makes them useful for applications to other fields, such as shock capturing computations or image processing. We start proving the approximation capabilities of an algorithm to detect the presence of isolated singularities, and then we address the approximation order attained by the mentioned interpolation procedure. For certain nonuniform grids with a maximum spacing between nodes $h$ below a critical value $h_c$, the optimal approximation order is recovered, as it happens for uniformly smooth functions \cite{ACDD}.