European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS7: Robust discretizations for coupled elliptic/parabolic equations (Part 3)
Chair: Florin Radu
25 mins
Interfaces in porous-media systems are everywhere: evaporation and salt precipitation at the atmosphere-unsaturated interface
Katharina Heck, Kilian Weishaupt, Edward 'Ned' Coltman, Melanie Lipp
Abstract: Fluid-filled porous media are ubiquitous in many natural and technical systems. However, our understanding of porous media continues to have substantial weaknesses, and great challenges for science, research, engineering and applications remain. Most of these challenges arise from interfaces: to a large extent, they control the flow and transport of fluids, solutes, particles and heat within porous media. These interfaces include fluid-fluid, fluid-structure and structure-structure interfaces, and their influences on the effective behaviour of porous-media systems occur on different scales in space and time. The relevance of interfaces is especially pronounced in complex coupled processes such as infiltration, multi-phase flow, evaporation from porous media, chemical reactions that change the pore structure, deposition of solids within the pore space and fracture propagation through porous media. The focus of this presentation will be on the atmosphere and unsaturated flow processes affected by turbulence. The results will be discussed and compared with experimental measurements on different scales. We will present various examples illustrating the influence of soil-moisture processes in the subsurface on the groundwater budget and quality.
25 mins
Dynamic and weighted stabilizations of the L-scheme applied to a phase-field model for fracture propagation
Christian Engwer, Iuliu Sorin Pop, Thomas Wick
Abstract: We consider a phase-field fracture propagation model, which consists of two (nonlinear) coupled partial differential equations. The first equation describes the displacement evolution, and the second is a smoothed indicator variable, describing the crack position. We propose an iterative scheme, the so-called $L$-scheme, with a dynamic update of the stabilization parameters during the iterations. Our algorithmic improvements are substantiated with two numerical tests. The dynamic adjustments of the stabilization parameters lead to a significant reduction of iteration numbers in comparison to constant stabilization values.
25 mins
A domain decomposition method for two-phase flow models in heterogeneous porous media
Kundan Kumar, Florian List, Stephan Lunowa, Koondanibha Mitra, Iuliu Sorin Pop, Florin Adrian Radu
Abstract: Two-phase porous media flows appear in several fields of highest societal relevance, such as environmental engineering, energy resources management, or oil recovery. The underlying mathematical models can be expressed as systems coupling evolution and elliptic equations, having a nonlinear and possibly degenerate character. In many of the situations of practical relevance, the models are defined in heterogeneous or fractured media, involving homogeneous blocks. For coupling the models valid inside each homogeneous block, one needs to define appropriate conditions at the interface between two adjacent blocks. Next to the continuity of the normal fluxes of each phase, one assumes a certain relationship between the corresponding phase pressures. Whereas for standard models and if entry pressure effects are disregarded, this leads to the continuity of the phase pressures, for models involving dynamic capillarity and entry pressure effects one ends up with nonlinear transmission conditions. We first motivate the use of equilibrium or non-equilibrium of models and discuss the interface conditions for different model concepts. Then we propose a domain decomposition method combined with a linearization approach for solving coupled models of different types, defined in porous media with block type heterogeneities. The approach builds on the Schwarz non-overlapping decomposition scheme and on an L-type iterative linearization method.