Numerical recipes in FORTRAN (2nd ed.): the art of scientific computing
Numerical recipes in FORTRAN (2nd ed.): the art of scientific computing
SIAM Journal on Numerical Analysis
The Finite Ray Element Method for the Helmholtz Equation of Scattering: First Numerical Experiments
The Finite Ray Element Method for the Helmholtz Equation of Scattering: First Numerical Experiments
On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation
SIAM Journal on Numerical Analysis
A collocation method for high-frequency scattering by convex polygons
Journal of Computational and Applied Mathematics
A Galerkin Boundary Element Method for High Frequency Scattering by Convex Polygons
SIAM Journal on Numerical Analysis
A Sparse Discretization for Integral Equation Formulations of High Frequency Scattering Problems
SIAM Journal on Scientific Computing
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There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.