A Numerical Coordinate Transformation for Efficient Evaluation of Oscillatory Integrals over Wave Boundary Elements

When the Partition of Unity Method is applied to a discretised integral equation form of the Helmholtz operator, the computational cost is dominated by the evaluation of highly oscillatory integrals over discretisations. This paper presents a new numerical approach that is based on a coordinate transformation to a secondary discretisation that is aligned with an approximation to the curvilinear coordinate in which oscillation is taking place. This transforms a 2D surface integral into an equivalent set of 1D integrals that may be evaluated cheaply yet accurately. A method is proposed for dealing efficiently with difficulties that occurred at the edges of elements in previous work.