ALEX 2018 Workshop: Abstracts

Fractals, homogenization, and multigrid

Martin Heida${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$, Ralf Kornhuber${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$, Joscha Podlesny${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$, and Harry Yserentant${}^{\mathrm{(}\mathrm{3}\mathrm{)}}$

(1) Weierstraß-Institut Berlin (Germany)

(2) Institute of Mathematics, Free University of Berlin (Germany)

(3) Institute of Mathematics, Technichal University of Berlin (Germany)

The fractal perspective on spatial self-similarity of geological structures has already quite a history, cf., e.g., [6]. Though simulation is obviously needed to overcome observational gaps, current activities in mathematical modelling and numerical approximation of fault network behavior seem to be rare and limited to single faults or simple fault geometries (see, e.g., [5] and the references cited therein).

On this background, we consider a scalar elliptic model problem with jump conditions on a sequence of multiscale networks of interfaces and suggest a new concept, called fractal homogenization, to derive and analyze an associated asymptotic limit problem [2]. The resulting “fractal” solution space is characterized in terms of generalized jumps and gradients, and we prove continuous embeddings into $L^2$ and $H^s$, on suitable assumptions on the geometry of the multiscale interface network.

We also present a numerical homogenization strategy in the spirit of [3, 4] which can be regarded as a re-interpretation of well-established concepts for multiscale problems [1] in terms of multigrid methods. We analyze the convergence properties of corresponding iterative solvers as well as the discretization error of corresponding discretization schemes by investigating the stability and approximation properties of certain quasi-projections.

Our theoretical findings are illustrated by numerical computations.

Acknowledgments: This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 ”Scaling Cascades in Complex Systems”, Project C05 ”Effective models for interfaces with many scales” and Project B01 ”Fault networks and scaling properties of deformation accumulation”.