A comparison of high-order and plane wave enriched boundary element basis functions for Helmholtz problems

Abstract

When undertaking a numerical solution of Helmholtz problems using the Boundary Element Method (BEM) it is common to employ low-order Lagrange polynomials, or more recently Non-Uniform Rational B-Splines (NURBS), as basis functions. A popular alternative for high frequency problems is to use an enriched basis, such as the plane wave basis used in the Partition of Unity Boundary Element Method (PUBEM). To the authors’ knowledge there is yet to be a thorough quantification of the numerical error incurred as a result of employing high-order NURBS and Lagrange polynomials for wave-based problems in a BEM setting. This is the focus of the current work, along with comparison of the results against PUBEM. The results show expected improvements in the convergence rates of a Lagrange or NURBS scheme as the order of the basis functions is increased, with the NURBS basis slightly outperforming the Lagrange basis. High-order Lagrange and NURBS formulations can compare favourably against PUBEM for certain cases. In addition, the recently observed pollution effect in BEM is studied for a travelling wave in a duct and the numerical dispersion presented for all three sets of basis functions.