arXiv subjects

Markus Gahn

Publications and source records attributed to Markus Gahn.

23 records · Page 2Linked to original sources

arXiv2021

Homogenization of a two-phase problem with nonlinear dynamic Wentzell-interface condition for connected-disconnected porous media

We investigate a reaction-diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $ε$. On the interface between the components we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk-domains are given by a reaction-diffusion equation for the traces of the bulk-solutions, including nonlinear reaction-kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $ε\to 0$ and derive macroscopic models, where we need homogenization results for surface diffusion. To cope with the nonlinear terms we derive strong two-scale results.

math.AP↗

arXiv2021

Homogenization of a reaction-diffusion-advection problem in an evolving micro-domain and including nonlinear boundary conditions

We consider a reaction-diffusion-advection problem in a perforated medium, with nonlinear reactions in the bulk and at the microscopic boundary, and low diffusion scaling. The microstructure changes in time; the microstructural evolution is known a priori. The aim of the paper is the rigorous derivation of a homogenized model. We use appropriately scaled function spaces, which allow us to show compactness results, especially regarding the time-derivative and we prove strong two-scale compactness results of Kolmogorov-Simon-type, which allow to pass to the limit in the nonlinear terms. The derived macroscopic model depends on the micro- and the macro-variable, and the evolution of the underlying microstructure is approximated by time- and space-dependent reference elements.

math.AP↗

arXiv2021

Two-scale tools for homogenization and dimension reduction of perforated thin layers: Extensions, Korn-inequalities, and two-scale compactness of scale-dependent sets in Sobolev spaces

This investigation develops basic methods for the multi-scale analysis for problems in thin porous layers. More precisely, we provide tools for the homogenization in case of "tangentially" periodic structures and dimensional reduction letting the layer thickness tend to zero proportional to the scale parameter $ε$. A crucial point is the identification of scale limits of functions $v_ε$ in subsets of function spaces characterized by uniform a priori estimates with respect to $ε$, arising for solutions of differential equations in heterogeneous media with thin layers, e.g., of a Navier-Stokes system, models in linear elasticity, or problems with fluid-structure interaction. Often in problems from continuum mechanics, in a first step, the symmetric gradients of arising vector fields can be controlled and Korn's inequality in porous layers is required to estimate the gradients, such that crucial constants do not depend on $ε$. Controllable pore-filling extension are constructed and, thus, the analysis is reduced to a fixed basic domain. The proof of the required Korn-inequalities for porous thin layers, formulated with respect to $L^p$-spaces, is based on these constructions. Also, the investigation of compactness with respect to two-scale convergence and the characterization of the scale limits is strongly based on the extension theorem and the Korn-inequalities. To illustrate the range of applications of the developed analytic multiscale method a semi-linear elastic wave equation in a thin periodically perforated layer with an inhomogeneous Neumann boundary condition on the surface of the elastic substructure is treated and an homogenized, reduced system is derived.

math.AP↗

arXiv2021

Partial differential equations on hypergraphs and networks of surfaces: derivation and hybrid discretizations

We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces---generalized by the term hypergraphs. To this end, we consider PDEs on hypergraphs as singular limits of PDEs in networks of thin domains (such as fault planes, pipes, etc.), and we observe that (mixed) hybrid formulations offer useful tools to formulate such PDEs. Thus, our numerical framework is based on hybrid finite element methods (in particular, the class of hybrid discontinuous Galerkin methods).

math.NA↗

arXiv2020

Effective transmission conditions for reaction-diffusion processes in domains separated by thin channels

We consider a reaction--diffusion equation in a domain consisting of two bulk regions connected via small channels periodically distributed within a thin layer. The height and the thickness of the channels are of order $ε$, and the equation inside the layer depends on the parameter $ε$ and an additional parameter $γ\in [-1,1)$, which describes the size of the diffusion in the layer. We derive effective models for the limit $ε\to 0 $, when the channel-domain is replaced by an interface between the two bulk-domains.

math.AP↗