Numerical Homogenization of Fractal Interface Problems

TL;DR

This paper develops a numerical homogenization method for fractal interface problems, enabling stable multiscale discretizations and solvers that are effective despite complex fractal geometries.

Contribution

It introduces projection operators with stability and approximation properties tailored for fractal geometries, facilitating multiscale discretizations and iterative solvers.

Findings

Constructed projection operators with stability and approximation properties.

Developed multiscale discretizations for fractal interface problems.

Achieved scale-independent convergence in iterative solvers.

Abstract

We consider the numerical homogenization of a class of fractal elliptic interface problems inspired by related mechanical contact problems from the geosciences. A particular feature is that the solution space depends on the actual fractal geometry. Our main results concern the construction of projection operators with suitable stability and approximation properties. The existence of such projections then allows for the application of existing concepts from localized orthogonal decomposition (LOD) and successive subspace correction to construct first multiscale discretizations and iterative algebraic solvers with scale-independent convergence behavior for this class of problems.