Applied Numerical Mathematics 44 (2003) 507–526 www.elsevier.com/locate/apnum Variational bivariate interpolating splines with positivity constraints A. Kouibia ∗ , M. Pasadas Department of Applied Mathematics, Faculty of Sciences, University of Granada, 18071 Granada, Spain Abstract This paper deals with a shape preserving method of interpolating positive data at points of the plane in R 2 . We formulate a problem in order to define a positive interpolation variational spline in the Sobolev space H m (Ω), where Ω is a non-empty bounded set of R 2 , by minimizing the semi-norm of order m, and we discrete such problem in a finite element space. An algorithm allows us to compute the resulting function. Some convergence theorems are established. The error is of the order O(1/N) m when N tends to +∞, being N the number of the interpolating points. Some numerical and graphical examples are given in order to test the validity of this method.  2002 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Shape preserving interpolation; Approximation of surfaces; Splines 1. Introduction Interpolation problems involving convex splines have recently received considerable attention, especially the bi-dimensional case. For a given surface with a certain shape, it is very unlikely that the approximating surface preserves the same shape. This is why we cannot find enough references interested in studying this type of problem. Then again, classical methods, where the polynomial spline functions are most widely used, usually ignore such conditions and thus yield solutions exhibiting undesirable inflections or oscillations. In early papers concerned with the bi-dimensional case, Utreras [13] considers the problem of interpolating positive data by means of positive thin plate splines at scattered data points of the plane R 2 . Likewise, in [7] we preserve the shape of data in such a way that in each real interval I where the data are positive, monotone or convex the solution of the method should have the same property. * Corresponding author. E-mail addresses: kouibia@ugr.es (A. Kouibia), mpasadas@ugr.es (M. Pasadas). 0168-9274/02/$30.00  2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0168-9274(02)00174-5