Abstract

Our focus is on the accurate numerical solution of PDEs and related spectral problems in domains with highly non-smooth (e.g. fractal) boundary. For such problems, an obvious approach is to first replace the domain by a smoother “prefractal” approximation, then apply a classical discretization, such as a conforming finite element method (FEM) on a simplicial mesh. However, a drawback of this approach is that to minimise the effect of geometry-related approximation errors one needs a highly accurate prefractal approximation, which generally leads to extremely complicated FEM meshes with a large number of elements. In this talk we present a novel class of geometry-conforming FEM -type discretizations in which the fractal domain is meshed exactly using a finite number of non-simplicial elements which themselves have fractal boundary. As we shall demonstrate, while such meshes cannot be used in the context of classical conforming FEM , they can be successfully applied in the context of both (i) discontinuous Galerkin FEM and (ii) integral equation methods, offering significantly more efficient approximation than comparable prefractal-based methods.   This is joint work with Andrea Moiola (Pavia), Sergio Gomez (Milano Bicocca), Joshua Bannister (UCL) and Andrew Gibbs (UCL).