The spheroidal wave functions, which are the solutions to the Helmholtz equation in spheroidal coordinates, are notoriously difficult to compute. Because of this, practically no programming language comes equipped with the means to compute them. This makes problems that require their use hard to tackle. We have developed computational software for calculating these special functions. Our software is called spheroidal and includes several novel features, such as: using arbitrary precision arithmetic; adaptively choosing the number of expansion coefficients to compute and use; and using the Wronskian to choose from several different methods for computing the spheroidal radial functions to improve their accuracy. There are two types of spheroidal wave functions: the prolate kind when prolate spheroidal coordinates are used; and the oblate kind when oblate spheroidal coordinate are used. In this paper, we describe both, methods for computing them, and our software. We have made our software freely available on our webpage.

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Many boundary element integral equation kernels are based on the Green's functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, which may be accelerated via the fast multipole method. There are advantages to Galerkin formulations for such integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires each entry in the system matrix to be created via the computation of a double surface integral over one or more pairs of triangles. There are a number of semi-analytical methods to treat these integrals, which all have some issues, and are discussed in this paper. We present novel methods to compute all the integrals that arise in Galerkin formulations involving kernels based on the Laplace and Helmholtz Green's functions to any specified accuracy. Integrals involving completely geometrically separated triangles are non-singular and are computed using a technique based on spherical harmonics and multipole expansions and translations, which results in the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. Example results are presented, and the developed software is available as open source.

Analytical solutions to acoustic scattering problems involving nonspherical shapes, such as spheroids and disks, have long been known and have many applications. However, these solutions require special functions that are not easily computable. For this reason, their asymptotic forms are typically used since they are more readily available. We explore these solutions and provide computational software for calculating their nonasymptotic forms, which are accurate over a wide range of frequencies and distances. This software, which runs in MATLAB, computes the solutions to acoustic scattering problems involving spheroids and disks by semi-analytical means, and is freely available from our webpage.

The indirect boundary element method for the Helmholtz equation in three dimensions is of great interest and practical value for many problems in acoustics as it is capable of treating infinitely thin plates and allows coupling of interior and exterior scattering problems. In the present paper, we provide a new approach for treatment of boundary integrals, including hypersingular, singular, and nearly singular integrals via analytical expressions for generic boundary conditions on the both sides of the surface. The fast multipole accelerated boundary element solver in Gumerov and Duraiswami (2009) is extended to incorporate the indirect formulation. The new formulation is compared with the analytical solution of scattering off a disk. Previous authors have not provided such comparisons for an extended range of frequencies. The performance of the method and its scalability are investigated. It is demonstrated that problems with millions of boundary elements can be solved efficiently on a personal computer using the present method.