Design of quadrature rules for Müntz and Müntz‐logarithmic polynomials using monomial transformation

Abstract

SUMMARYA method for constructing the exact quadratures for Müntz and Müntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for Müntz polynomials.

Singular Higher Order Divergence-Conforming Bases of Additive Kind and Moments Method Applications to 3D Sharp-Wedge Structures

“…These effects are in general expected to be negligible in the far-field because: a) the radiation integral-operator cuts off the highest spatial-frequency components of the charge and current densities arising from the singular behavior of these quantities in the wedge neighborhood; b) the MoM-Galerkin method used to approximately evaluate the surface charge and current densities is variational (see [24,Ch. 1.8], [25], [38] and references therein). At any rate, in the attempt to quantitatively clarify these effects, Fig.…”

“…However, as we will see in the following, the major drawback of these lowest order bases with respect to those of Table I is that either some of their singular basis functions contain a fractional term, as it happens when using the A-type potentials on edge-singularity triangles, or that they contain some other spurious higher order term for the other elements (i.e., the quadrilateral and the triangular vertex-singularity element) in case one uses the B-type potentials. In MoM applications, the presence of these undesired terms considerably complicates the numerical evaluation of the required integrals to the desired accuracy [38].…”

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“…The method can be further extended to include high order basis functions [10] and singular basis functions [11][12] where special quadratures are needed [13].…”