Filling polygonal holes with minimal energy surfaces on Powell–Sabin type triangulations

In this paper we present two different methods for filling in a hole in an explicit 3D surface, defined by a smooth function ff in a part of a polygonal domain D⊂R2D⊂R2. We obtain the final reconstructed surface over the whole domain DD. We do the filling in two different ways: discontinuous and continuous. In the discontinuous case, we fill the hole with a function in a Powell–Sabin spline space that minimizes a linear combination of the usual seminorms in an adequate Sobolev space, and approximates (in the least squares sense) the values of ff and those of its normal derivatives at an adequate set of points. In the continuous case, we will first replace ff outside the hole by a smoothing bivariate spline sfsf, and then we fill the hole also with a Powell–Sabin spline minimizing a linear combination of given seminorms. In both cases, we obtain existence and uniqueness of solutions and we present some graphical examples, and, in the continuous case, we also give a local convergence result.