Speaker
David Hewett, University College London
Abstract
We consider acoustic scattering problems for fractal inhomogeneities, i.e. acoustic transmission problems involving fractal interfaces. Such problems are a simple model for certain electromagnetic scattering problems arising in atmospheric physics, involving the scattering of the sun’s radiation by complex ice crystal aggregates in clouds. For our analysis and numerical approximation we use a classical Lippmann-Schwinger volume integral equation formulation. Approximation of the fractal inhomogeneity by a smoother “prefractal” inhomogeneity and application of a standard volume integral equation solver can lead to sub-optimal convergence, sometimes referred to as “stair-casing effects”. To avoid this, our method works with a discretization of the true fractal inhomogeneity using a “fractal mesh” comprising self-similar fractal elements which fully preserve the fractal geometry of the inteface. Numerical quadrature for the singular integrals that arise is performed with the help of singularity subtraction combined with tricks that exploit the self-similarity of the elements. Our piecewise constant method gives optimal (O(h^2)) superconvergence of the scattered field and far-field pattern, and is supported by a fully discrete error analysis and numerical results for examples including the Koch snowflake. This is joint work with Joshua Bannister (UCL) and Andrew Gibbs (UCL).