Polynomial Interpolation of Function Averages on Interval Segments

Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyse fundamental mathematical properties of this problem as existence, uniqueness and numerical conditioning of its solution. We will provide concrete conditions for unisolvence, explicit Lagrange-type basis systems for its representation, and a numerical method for its solution. To study the numerical conditioning, we will provide concrete bounds of the Lebesgue constant in a few distinguished cases.Comment: 20 pages, 5 figure

NEFEM for EULER equations

An improvement of the classical finite element method is proposed in [1], the NURBS-Enhanced Finite Element Method (NEFEM). It considers an exact representation of the geometry by means of the usual CAD description of the boundary with Non-Uniform Rational B-Splines (NURBS). For elements not intersecting the boundary, a standard finite element interpolation and numerical integration is used. Specifically designed piecewise polynomial interpolation and numerical integration are proposed for those finite elements intersecting the NURBS boundary. In [2] a numerical example involving an electromagnetic scattering application, is used in order to demonstrate the applicability and behavior of the proposed methodology. The results are encouraging and show that the NEFEM is more accurate than the corresponding isoparametric finite elements, using a Discontinuous Galerkin (DG) formulation. Recent studies also demonstrate that, for a desired precision, the NEFEM is also more efficient in terms of number of degrees of freedom, and in terms of CPU time. In the present work the NEFEM is reviewed and applied to the solution of the Euler equations of a compressible inviscid fluid. This set of hyperbolic equations represents a more challenging application for the NEFEM because the nonlinearity of the hyperbolic system and the sensitivity of DG formulations to the imposition of the wall boundary condition in curved domains.Peer Reviewe