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).
