Non-oscillatory butterfly-type interpolation on triangular meshes

This paper proposes and analyses a non-oscillatory interpolatory subdivision scheme for data on regular triangular grids, a non-linear analogue of the well-known butterfly subdivision scheme. The scheme is obtained, first, by re-interpreting the butterfly refinement rule as a linear combination of ‘smaller'linear rules and, then, by introducing a particular type of non-linear average in place of the linear ones. We end up with a non-linear interpolatory subdivision scheme that, applied to discrete data with large gradients, shows no Gibbs-like oscillatory phenomenon, while behaves similarly to the butterfly scheme when applied to smooth data. Convergence, reproduction and approximation properties of the proposed scheme are investigated and several numerical examples are discussed.