A Convergence Analysis of the Multipole Expansion Method
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The multipole expansion method (MEM) is a spatial discretization technique that is widely used in applications that feature scattering of waves from multiple spheres and circular cylinders. Moreover, it also serves as a key component in several other numerical methods in which scattering computations involving arbitrarily shaped objects are accelerated by enclosing the objects in artificial spheres or cylinders. A fundamental question is that of how fast the approximation error of the MEM converges to zero as the truncation number goes to infinity. Despite the fact that the MEM was introduced in 1913, and has been in widespread usage as a numerical technique since as far back as 1955, to the best of the authors' knowledge, a precise, quantitative characterization of the asymptotic rate of convergence of the MEM has not been obtained. In this work, we finally provide a resolution to this issue for the two-dimensional case. We begin by deriving bounds which are tight as long as the cylinders are not too close together. When some cylinders are, in fact, in close proximity to one another, these bounds become pessimistic. To obtain a more accurate characterization of the convergence in this regime, we formulate a first-order scattering approximation and derive its rate of convergence. Numerical simulations show that this approximation provides a far more accurate estimate of the convergence in the closely spaced regime than the aforementioned bounds. Our results establish MEM convergence rates that hold for all boundary conditions and boundary integral equation solution representations.
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