Smooth bivariate shape-preserving cubic spline approximation

Given a piece-wise linear function defined on a type I uniform triangulation we construct a new partition and define a smooth cubic spline that approximates the linear surface and preserves its shape. The key piece is a new macro-element that has the ability to combine six independent gradients coming together at an interior vertex in a smooth yet shape preserving fashion. The shape of the resulting spline surface follows local changes in the shape of the piece-wise linear interpolant without overshooting. We prove that convexity, positivity and monotonicity of the spline depend on the local data only. Computational scheme for Bernstein-Bezier spline coefficients is local and fast. Numerical examples highlight unique shape-preserving properties of the spline. (C) 2016 Elsevier B.V. All rights reserved.