Polynomial Histopolation, Superconvergent Degrees of Freedom, and Pseudospectral Discrete Hodge Operators

We show that, given a histogram with n bins—possibly non-contiguous or consisting of single points—there exists a unique polynomial histopolant (polynomial of order n with bin averages equal to the histogram’s). We also present histopolating cardinal functions for histograms with contiguous bins. The Hodge star operator from the theory of differential forms is the mapping which puts kand (d − k)-forms on a d-manifold into one-to-one correspondence. For k = d, the star operator is a bijective mapping between d-forms (“densities”) and 0-forms (“scalar functions”); a discrete analog is an operator which, given the integrals of a function over n cells, reconstructs (approximately) the values of the function at n nodes. We use histopolants to construct discrete Hodge star operators based on univariate polynomials. More generally, consider a linear operator which converts n degrees of freedom associated with univariate functions—integrals, averages and/or point values of a function and its derivatives, expansion coefficients with respect to polynomials bases, etc.—to n nodal values. Suppose that this operator is exact on Pn, the space of polynomials of order n. Is it possible to choose the nodes so that the operator is actually exact on Pn+1? We characterize the operators for which this is possible, and show that the “optimal” nodes are unique if they exist. We then give explicit formulas for the optimal nodes which correspond to star operators based on histopolation. For example, discrete Hodge star operators based on dual Legendre grids are superconvergent. Numerical tests confirm the superconvergence and show that the matrix representations of discrete star operators are close to the identity when dual Legendre or Chebyshev grids are used.