Transfinite Elements Using Bernstein Polynomials

Transfinite interpolation, originally proposed in the early 1970s as a global interpolation method, was first implemented using Lagrange polynomials and cubic Hermite splines. While initially developed for computer-aided geometric design (CAGD), the method also found application in global finite element analysis. With the advent of isogeometric analysis (IGA), Bernstein-B\'ezier polynomials have increasingly replaced Lagrange polynomials, particularly in conjunction with tensor-product B-splines and non-uniform rational B-splines (NURBS). Despite its early promise, transfinite interpolation has seen limited adoption in modern CAD/CAE workflows, primarily due to its mathematical complexity--especially when blending polynomials of different degrees. In this context, the present study revisits the classical formulation of transfinite interpolation and demonstrates that, in four broad classes, Lagrange polynomials can be systematically replaced by Bernstein polynomials in a one-to-one manner. This replacement yields a robust dual set of basis functions with improved numerical properties. A key advantage of Bernstein polynomials lies in their natural compatibility with weighted formulations, enabling the accurate representation of conic sections and quadrics--scenarios where IGA methods are particularly effective. The proposed methodology is validated through its application to a boundary-value problem governed by the Laplace equation, confirming both its feasibility and accuracy.