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This thesis deals with mathematical modeling, analysis, and numerical realization of microaggregates in soils. These microaggregates have the size of a few hundred micrometers and can be understood as the fundamental building units of soil. Thus, understanding their dynamically evolving, three-dimensional structure is crucial for modeling and interpreting many soil parameters such as diffusivities and flow paths that come into play in CO2-sequestration or oil recovery scenarios.
Among others, the following aspects of the formation of microaggregates should be incorporated into a mathematical model and investigated in more detail: the spatial heterogeneity of the temporally evolving structure of microaggregates and the different processes that take place on different scales — temporal and spatial — within the so-called micro-scale itself. This work aims at formulating a process-based pore-scale model, where all chemical species are measured in concentrations. That is, we have a continuous model for reactive transport mainly in terms of partial differential equations (PDEs) with algebraic constraints. This continuous model is defined on a discrete and discretely moving domain whose geometry changes according to the rules of a cellular automaton method (CAM). These rules describe the restructuring of the porous matrix, growth and decay of biomass, and the resulting topological changes of a wetting fluid and a gas phase. The cellular automaton rules additionally imply stochastic aspects that are important on the pore-scale.
Moreover, effects and knowledge deduced from the model are transfered to scales which are more relevant for applications. The quality of these averaged models is of general interest, since simulations for the field-scale that resolve the pore-scale are not applicable for economical reasons. Thus, this book compares parameterizations of diffusivities with mathematically rigorous results and gives suggestions to improve the formulas that can be found in the literature.
The discrete movement of the microaggregates’ geometry at the micro-scale poses mathematical problems. The following question arises: Can the averaged quantities deduced from the pore-scale really be used for models on other scales or are the impacts of the artificial temporal jumps too detrimental for the solutions on other scales to be accurate? In the following, this problem is also dealt with, and the reliability of the obtained parameters is underlined.
Last but not least, it is imperative to apply a proper numerical method to implement the model in silico. The local discontinuous Galerkin (LDG) method seems to be suitable for this task, since it is locally mass-conservative and is stable for discontinuous data — that might, for example, originate from the discrete movement of the geometry or from the sharp boundaries between the different phases. Additionally, this method has no problems with complicated transfer conditions. These aspects are demonstrated in a mathematically rigorous way, and the method is improved upon by reducing the linear system of equations resulting from the discretization. This is a real enhancement, since it does not diminish the order of convergence but decreases the computational costs.

We consider the dynamics of dilute electrolytes and of dissolved charged particles within a periodic porous medium at the pore scale, which is described by the non-stationary Stokes–Nernst–Planck–Poisson (SNPP) system.
Since simulations that resolve the geometry of the solid matrix at the pore scale are not feasible in practice, a major interest lies in the quality assessment of corresponding averaged models. Depending on the chosen scaling, the different averaged models under investigation reasonably describe to a greater or lesser extent the effective macroscopic behavior of the phenomena considered. The underlying partial differential equations include effective tensors, the closed-form expression of which is provided by averaging of the solutions of auxiliary problems. These so-called cell problems are defined on small domains reflecting the periodic geometry of the solid matrix.
The main objectives are both the qualitative and the quantitative investigation of homogenization processes by means of an extensive numerical study, i.e., of the convergence properties of the SNPP systems for vanishing microstructure. To this end, numerical schemes are proposed that are capable of solving accurately and efficiently the non-stationary, fully coupled/nonlinear SNPP system and also the corresponding averaged systems. The discretization is performed fully implicitly in time, while using mixed finite elements in two space dimensions, which are locally mass conservative with respect to the concentration of charged particles. The schemes are of optimal order in the discretization parameters, which is demonstrated numerically and also shown rigorously by an a priori error estimate for the overall discretization error.
Subsequently, the thesis proceeds with the numerical realization of an extension to the SNPP system allowing for attachment and detachment processes on the surface of the considered locally periodic solid matrix. The resulting evolving microstructure has an impact on the liquid flow and thus consequently on the solute transport. The corresponding two-scale model, which contains these inter-scale dependencies, is approached numerically using mixed finite elements on both scales. Simulations illustrate the interplay between solute transport, evolving microstructure, and liquid flow.

We develop a numerical framework for efficiently
simulating partially miscible multiphase multicomponent flow in porous media with general chemical reactions,
modeled by a system of coupled and strongly nonlinear partial differential equations, ordinary differential equations, and algebraic equations.
The equations are transformed with the help of a reduction
scheme which preserves the model and allows the unknown equilibrium reaction rates to be eliminated. In the transformed system, the algebraic equations resulting from the chemical
equilibria are eliminated in terms of a nonlinear, implicitly defined resolution
function. Hereby, the equilibrium conditions consisting of equations and inequality constraints are formulated as a complementarity problem and rewritten as
algebraic equations using the minimum function. The existence of the resolution
function is established by using the connection of the nonlinear system of algebraic equations to a constrained minimization problem based on a modified Gibbs
functional, and numerical strategies to evaluate it numerically are presented. Using a global implicit approach, the nonlinear systems that remain to be solved
in each time step are treated with the semismooth Newton method. With the help of the
resolution function, we are ready to define a persistent set of primary variables that
are valid in either phase state and for an arbitrary mineral assemblage. Different numerical tests related to hydrogen migration in deep geological repositories of radioactive waste and to CO2 sequestration show that our solver provides accurate numerical results, and that it is capable of
handling the strongly nonlinear coupling of flow, transport, chemical reactions,
and mass transfer across the phases.
The second part of this work is concerned with the analysis of robust mixed hybrid finite
element methods of lowest order for advection–diffusion problems. In the approach studied
here, the classical scheme is extended with the help of the
Lagrange multipliers associated with the hybrid problem formulation. More precisely, it relies
on the fact that the Lagrange multipliers represent approximations of the scalar
unknown on the interelement boundaries and uses them in the approximation of
the advective fluxes. Using techniques from the a posteriori error analysis, we are
able to establish optimal order convergence for a new class of methods including
the standard method, a partial upwind method, and a full upwind method. The
advantage of the specific choice of the upwind weights is the fact that the method
remains local. In this case, static condensation can be employed, whereas the standard upwind-mixed method requires information from neighbor cells such that
static condensation is not applicable.
Concerning approximations with the BDM1 mixed finite element, we show that
the use of Lagrange multipliers in the discretization of the advective term provides optimal second order convergence for the total flux variable in the L2 norm, whereas the standard mixed method
is known to be of suboptimal first order accuracy only.

In the first part of this thesis, we derive a thermodynamically consistent mathematical model for electrolyte solutions within the framework of nonequilibrium thermodynamics. Furthermore, by applying suitable simplifying assumptions, we show that this model reduces to the well-known and widely used family of Poisson–Nernst–Planck systems. Hence, for these classical models we provide a thermodynamical verification, we clearly reveal their limitations, and we present thermodynamically consistent extensions.
Moreover, we are concerned with the formation of particle clusters in electrolyte solutions. This ubiquitous and important process is known as coagulation. We derive a model for coagulation in electrolyte solutions, which is based on the Poisson-Nernst-Planck system. We connect this continuum mechanical system with the atomistic DLVO-theory. Thereby, we obtain a micro-macro model, which accounts for the short-range atomistic picture, the long-range continuum mechanical picture, the energetic DVLO-picture, and the kinetic picture of coagulation.
In the second part of this thesis, we present results for the global existence and uniqueness. We prove the global existence and uniqueness of weak solutions of the Darcy–Poisson–Nernst–Planck system, which is a field-scale model for electrolyte solutions in porous media. We consider multicomponent electrolyte solutions, which consist of a neutral solvent and charged solutes with arbitrary valencies.

The discontinuous Galerkin method for free surface and subsurface flows in geophysical applications
(2020)

Free surface flows and subsurface flows appear in a broad range of geophysical applications and in many environmental settings situations arise which even require the coupling of free surface and subsurface flows. Many of these application scenarios are characterized by large domain sizes and long simulation times. Hence, they need considerable amounts of computational work to achieve accurate solutions and the use of efficient algorithms and high performance computing resources to obtain results within a reasonable time frame is mandatory.
Discontinuous Galerkin methods are a class of numerical methods for solving differential equations that share characteristics with methods from the finite volume and finite element frameworks. They feature high approximation orders, offer a large degree of flexibility, and are well-suited for parallel computing.
This thesis consists of eight articles and an extended summary that describe the application of discontinuous Galerkin methods to mathematical models including free surface and subsurface flow scenarios with a strong focus on computational aspects. It covers discretization and implementation aspects, the parallelization of the method, and discrete stability analysis of the coupled model.