# Second French-German Workshop on Multiscale Problems

The Laboratoire de Mathématiques de Besançon organizes the French-German workshop on numerical methods for multi-scale problems.This is the second installment of the series initiated (virtually) in June 2020. The workshop will cover some recent advances in this research area (LOD, MsFEM, ...) and provide space for informal discussions.

The workshop will be held at the Espace Grammont (centre diocésain) in the center of Besançon

20, rue Mégevand

25000 Besançon

The venue is at the walking distance form the Besançon-Viotte railway station. It is also easily accessible by the tramway (stop *Révolution*) or the bus L3 (stop *8 Septembre*).

**Contact :**

**Confirmed participants :**

Grégoire Allaire (Paris)

Loic Balazi (Paris)

Rutger Biezemans (Paris)

Ralf Kornhuber (Berlin)

Claude Le Bris (Paris)

Frederic Legoll (Paris)

Alexei Lozinski (Besançon)

Pascal Omnès (Paris)

Daniel Peterseim (Augsburg)

**Preliminry program**** :**

**Wednesday 22/09 **

**15:00 Daniel Peterseim**,*Super-localization of elliptic multiscale problems*Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a*d*-dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter*H.*This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximate solution space. This talk presents a novel localization technique that enforces the super-exponential decay of the basis relative to*H.*This shows that basis functions with supports of width*O(H*|log*H|*^{(d-1)/d}*)*are sufficient to preserve the optimal algebraic rates of convergence in*H**O(H*|log*H|).*

Joint work with: Mortiz Hauck, https://arxiv.org/abs/2107.13211**16:30 Grégoire Allaire,***Analysis of a multiscale finite element method for convection-diffusion models*

**Thur****sday 2****3****/09 **

**9:00****Rutger Biezemans,***MsFEM for advection-diffusion problems*The MsFEM is a finite element (FE) approach that allows to solve partial differential equations (PDEs) with highly oscillatory coefficients on a coarse mesh by the use of problem-adapted basis functions. The MsFEM theory is mostly developed for symmetric elliptic problems. Additional difficulties arise in the numerical approximation of PDEs with dominating advection terms, due to the existence of steep boundary layers in the exact solution; naive FE approximation may lead to spurious oscillations even outside the boundary layer. Multiple stabilization methods exist today to adequately adapt FE methods to the resolution of advection-diffusion problems. In spite of various proposals to combine multi-scale and stabilization methods, a universally best method has not yet been identified.

In this talk, we will first present a number of existing stabilized multi-scale methods and next motivate the introduction of several new variants that are currently being investigated. Differences in the performance of the various methods will be illustrated with the help of numerical experiments.

slides**10:30****Ralf Ko****rn****huber,***T**owards numerical simulation of multiscale fault networks*

Stress accumulation and release in geological fault networks play a crucial role in earthquake dynamics. Motivated by the multiscale or even fractal character of such fault networks, we first consider a class of elliptic second order model problems with jump conditions on a sequence of networks of interfaces with a fractal asymptotic limit. We derive and analyze an associated (fractal) limit problem in order to quantify the effect of resolving the network only up to some finite number of interfaces. Numerical homogenization by local orthogonal decomposition and algebraic solution by multigrid methods is also discussed.

In the second part, we present a mathematical model for the deformation of a geological structure with non-intersecting faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and state-depending friction conditions along the common interfaces and undergoing large tangential displacements. We briefly sketch a numerical solution algorithm based on a Newmark discretization in time, decoupling of rate and state by a fixed point iteration, a mortar method in space and non-smooth multigrid. Numerical experiments illustrate the behavior of the model together with the efficiency and reliability of our solver.

slides

**14:00 Pascal Omnès,***Multi-scale finite elements and applications in fluid mechanics*slides**15:30****Frederic Legoll****,***On non-intrusive implementations of MsFEM approaches*slides

**Frid****ay 2****4****/09 **

**9:00 Alexei Lozinski,***On the homogenization of the stationary advection-diffusion equation*We shall present some rigourous and some formal results about the homogenization limits of the problem in the title at the regime dominated by advection. We shall also discuss how these results can be used to study the MsFEM approaches, at least in 1D.

slides