Convexity preserving interpolation and Powell-Sabin elements

Abstract This paper is concerned with constructing local convexity preserving bivariate interpolants. Piecewise linear upper and lower bounds for such interpolants are derived which induce certain partitions of the underlying domain. The fact that these partitions are data dependent and cannot be constructed by a local procedure reflects the difficulty in constructing local convexity preserving methods. However, simple additional conditions are established which ensure that Powell-Sabin elements provide C 1 convexity preserving interpolants.