High order interpolation of curves in the plane

We consider high order representations of curves in the plane. Through a series of complementary numerical examples, we show that classical interpolation is sometimes far from optimal in the sense of minimizing the interpolation error using a fixed number of degrees-of-freedom (the Kolmogorov n-width problem). We propose a new way of constructing a high order interpolant of a curve in the plane. The main ingredients are: a parametric representation of the curve, an implicit reparametrization of the curve, and choosing the internal interpolation points in such a way that the tangent vectors of the exact curve and the tangent vectors of the interpolant coincide at these points. Numerical results indicate that the proposed interpolant is close to optimal in the sense of minimizing the L 2-error between the exact curve and its interpolant. We also show that the error may decrease exponentially fast even for curves which are not analytic when regarded as a function in the classical setting. Finally, we use the proposed method to construct an improved representation of the boundary of a deformed quadrilateral domain. We show that the discretization error associated with solving the Poisson problem in the deformed domain may be significantly smaller than the error resulting from a standard approach.