Generalized high order interpolatory 1-form bases for computational electromagnetics

The explicit formulae for arbitrary order 1-form interpolatory bases developed by Graglia, Wilton and Peterson (1997) are based upon Silvester-Lagrange polynomials. It is well known that these polynomials have the potential to exhibit erratic behavior as the order p increases due to their rapidly increasing Lebesgue constants. In this paper we present computed Lebesgue constants for these 1-form bases and for new bases that utilize interpolation points based on the zeros of Chebyshev polynomials. These new bases have significantly smaller Lebesgue constants (near optimal), and we show by example that this directly affects the interpolation error. We show that for large p, the interpolation error for non-uniform bases can be orders of magnitude smaller than that of uniform bases. The procedure presented here is generic in the sense that any interpolatory polynomial can be used. This generality is achieved by constructing the 1-form basis function on a reference element and transforming to the actual element using appropriate transformation rules. The appropriate transformation rules are conceived by identifying the interpolation vectors as tangent vectors, the basis functions as 1-forms and the curl of the basis functions as 2-forms.