Joshua Bannister: Fractal and pre-fractal discretisation methods for the Lippmann-Schwinger equation

Speaker
Joshua Bannister, University College London

Abstract
In this talk, we discuss two distinct discretisation strategies for solving the Lippmann-Schwinger volume integral equation (LSE) on self-similar fractal sets by focusing on the Koch snowflake. The first, known as pre-fractal methods, involve approximating the snowflake by a sequence of smoother (e.g., Lipschitz) sets that converge in a suitable sense, thereby permitting the application of classical standard discretisation techniques. We show that a certain class of pre-fractal methods converges to the solution on the snowflake, but at a sub-optimal rate depending on the (non-integer) Hausdorff dimension of the snowflake’s boundary, a phenomenon that we refer to as “fractal staircasing”. The second method, referred to as the IFS method, exploits the geometric self-similarity of the snowflake by producing a mesh consisting of scaled and rotated copies of itself, on which a PWC Galerkin method can be defined. This method does not suffer from fractal staircasing and converges linearly to the solution on the snowflake. However, due to the low-order nature of the discretisation and the size of the resulting dense matrices, a naive implementation of such a method is practically unviable. To remedy this, we exploit the lattice structures in self-similar meshes using the FFT to compress the linear system and perform matrix-vector multiplication in log-linear complexity, accelerating the use of iterative methods.