A discontinuous Galerkin method with fractal elements

TL;DR

This paper develops a discontinuous Galerkin finite element method tailored for elliptic boundary value problems on fractal domains, specifically using fractal elements on the Koch snowflake boundary, with proven well-posedness and demonstrated numerical effectiveness.

Contribution

It introduces a novel DG-FEM approach directly on fractal geometries using fractal tilings, extending finite element methods to complex fractal boundaries.

Findings

Method is well-posed and quasi-optimal.

Numerical results confirm effectiveness for polynomial basis functions.

Applicable to eigenvalue problems on fractal domains.

Abstract

We formulate, analyse, and implement a discontinuous Galerkin finite element method (DG-FEM) for the approximation of the solution of an elliptic boundary value problem in a domain with fractal boundary. We consider the case of the Poisson equation in the Koch snowflake domain with zero Dirichlet boundary conditions, but our methodology can be generalised to other cases. Rather than first approximating the snowflake domain by a polygonal "prefractal" and then applying a standard DG-FEM on the prefractal, we define a DG-FEM on the snowflake itself, using a geometry-conforming mesh (a fractal tiling) consisting of fractal elements, each similar to the original snowflake. Fluxes across inter-element boundaries, which are fractal curves, are represented in a weak way by integrals over element subdomains. We show how, for local polynomial basis functions, these integrals can be evaluated exactly using the similarity of the elements. We prove well-posedness and quasi-optimality of the method, and provide a partial convergence analysis. We present numerical results for piecewise linear and piecewise quadratic basis functions, which demonstrate the effectiveness of the method. We also apply our method to the related Dirichlet eigenvalue problem.