Numerical solution of steady-state groundwater flow and solute transport problems: Discontinuous Galerkin based methods compared to the Streamline Diffusion approach

In this study, we consider the simulation of subsurface flow and solute transport processes in the stationary limit. In the convection-dominant case, the numerical solution of the transport problem may exhibit non-physical diffusion and under- and overshoots. For an interior penalty discontinuous Galerkin (DG) discretization, we present a $h$-adaptive refinement strategy and, alternatively, a new efficient approach for reducing numerical under- and overshoots using a diffusive $L^2$-projection. Furthermore, we illustrate an efficient way of solving the linear system arising from the DG discretization. In $2$-D and $3$-D examples, we compare the DG-based methods to the streamline diffusion approach with respect to computing time and their ability to resolve steep fronts

Efficient "black-box" multigrid solvers for convection-dominated problems

The main objective of this project is to develop a "black-box" multigrid preconditioner for the iterative solution of finite element discretisations of the convection-diffusion equation with dominant convection. This equation can be considered a stand alone scalar problem or as part of a more complex system of partial differential equations, such as the Navier-Stokes equations. The project will focus on the stand alone scalar problem. Multigrid is considered an optimal preconditioner for scalar elliptic problems. This strategy can also be used for convection-diffusion problems, however an appropriate robust smoother needs to be developed to achieve mesh-independent convergence. The focus of the thesis is on the development of such a smoother. In this context a novel smoother is developed referred to as truncated incomplete factorisation (tILU) smoother. In terms of computational complexity and memory requirements, the smoother is considerably less expensive than the standard ILU(0) smoother. At the same time, it exhibits the same robustness as ILU(0) with respect to the problem and discretisation parameters. The new smoother significantly outperforms the standard damped Jacobi smoother and is a competitor to the Gauss-Seidel smoother (and in a number of important cases tILU outperforms the Gauss-Seidel smoother). The new smoother depends on a single parameter (the truncation ratio). The project obtains a default value for this parameter and demonstrated the robust performance of the smoother on a broad range of problems. Therefore, the new smoothing method can be regarded as "black-box". Furthermore, the new smoother does not require any particular ordering of the nodes, which is a prerequisite for many robust smoothers developed for convection-dominated convection-diffusion problems. To test the effectiveness of the preconditioning methodology, we consider a number of model problems (in both 2D and 3D) including uniform and complex (recirculating) convection fields discretised by uniform, stretched and adaptively refined grids. The new multigrid preconditioner within block preconditioning of the Navier-Stokes equations was also tested. The numerical results gained during the investigation confirm that tILU is a scalable, robust smoother for both geometric and algebraic multigrid. Also, comprehensive tests show that the tILU smoother is a competitive method.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo

Numerical Simulation of Multiphase Flow in Fractured Porous Media

Fractures provide preferred paths for flow and transport in many porous media. They have a significant influence on process behavior in groundwater remediation, reservoir engineering and safety analysis for waste repositories. We present a finite volume method for the numerical solution of the multiphase flow equations in fractured porous media. The capillary pressure is treated by an extended capillary pressure interface condition. The method is fully coupled and fully implicit and employs a mixed-dimensional formulation with lower dimensional elements in the fractures. The method features unstructured grids, adaptive refinement and multigrid methods. It is implemented for twodimensional and threedimensional complex problems with several million unknowns. Additionally, a discontinuous Galerkin method for the groundwater flow equation and its multigrid treatment is presented

Coupling different discretizations for fluid structure interaction in a monolithic approach

In this thesis we present a monolithic coupling approach for the simulation of phenomena involving interacting fluid and structure using different discretizations for the subproblems. For many applications in fluid dynamics, the Finite Volume method is the first choice in simulation science. Likewise, for the simulation of structural mechanics the Finite Element method is one of the most, if not the most, popular discretization method. However, despite the advantages of these discretizations in their respective application domains, monolithic coupling schemes have so far been restricted to a single discretization for both subproblems. We present a fluid structure coupling scheme based on a mixed Finite Volume/Finite Element method that combines the benefits of these discretizations. An important challenge in coupling fluid and structure is the transfer of forces and velocities at the fluidstructure interface in a stable and efficient way. In our approach this is achieved by means of a fully implicit formulation, i.e., the transfer of forces and displacements is carried out in a common set of equations for fluid and structure. We assemble the two different discretizations for the fluid and structure subproblems as well as the coupling conditions for forces and displacements into a single large algebraic system. Since we simulate real world problems, as a consequence of the complexity of the considered geometries, we end up with algebraic systems with a large number of degrees of freedom. This necessitates the use of parallel solution techniques. Our work covers the design and implementation of the proposed heterogeneous monolithic coupling approach as well as the efficient solution of the arising large nonlinear systems on distributed memory supercomputers. We apply Newton’s method to linearize the fully implicit coupled nonlinear fluid structure interaction problem. The resulting linear system is solved with a Krylov subspace correction method. For the preconditioning of the iterative solver we propose the use of multilevel methods. Specifically, we study a multigrid as well as a two-level restricted additive Schwarz method. We illustrate the performance of our method on a benchmark example and compare the afore mentioned different preconditioning strategies for the parallel solution of the monolithic coupled system

Discontinuous Galerkin based Geostatistical Inversion of Stationary Flow and Transport Processes in Groundwater

The hydraulic conductivity of a confined aquifer is estimated using the quasi-linear geostatistical approach (QLGA), based on measurements of dependent quantities such as the hydraulic head or the arrival time of a tracer. This requires the solution of the steady-state groundwater flow and solute transport equations, which are coupled by Darcy's law. The standard Galerkin finite element method (FEM) for the flow equation combined with the streamline diffusion method (SDFEM) for the transport equation is widely used in the hydrogeologists' community. This work suggests to replace the first by the two-point flux cell-centered finite volume scheme (CCFV) and the latter by the Discontinuous Galerkin (DG) method. The convection-dominant case of solute (contaminant) transport in groundwater has always posed a special challenge to numerical schemes due to non-physical oscillations at steep fronts. The performance of the DG method is experimentally compared to the SDFEM approach with respect to numerical stability, accuracy and efficient solvability of the occurring linear systems. A novel method for the reduction of numerical under- and overshoots is presented as a very efficient alternative to local mesh refinement. The applicability and software-technical integration of the CCFV/DG combination into the large-scale inversion scheme mentioned above is realized. The high-resolution estimation of a synthetic hydraulic conductivity field in a 3-D real-world setting is simulated as a showcase on Linux high performance computing clusters. The setup in this showcase provides examples of realistic flow fields for which the solution of the convection-dominant transport problem by the streamline diffusion method fails

Multiskalen-Verfahren für Konvektions-Diffusions Probleme

In dieser Arbeit werden erstmalig über einen zur Nichtstandardform gehörenden Erzeugendensystem-Ansatz robuste Wavelet-basierte Multiskalen-Löser für allgemeine zweidimensionale stationäre Konvektions-Diffusions-Probleme entworfen und praktisch umgesetzt. Für Multiskalen-Verfahren, die lediglich direkte Unterraumzerlegungen verwenden, ist es im allgemeinen nicht mehr möglich, zugehörige Multiskalen-Glätter zu konstruieren, die im Grenzfall sehr starker Konvektion auf jeder Skala zu einem direkten Löser entarten. Als eine Möglichkeit zur Konstruktion robuster Multiskalen-Methoden bleibt die Wahl der Multiskalen-Zerlegungen selbst. Es ist sicherzustellen, dass man sowohl hinsichtlich der singulären Störung stabile Grobgitter- probleme als auch bezüglich der Maschenweite stabile Unterraum- zerlegungen erhält. Gleichzeitig muss der Aspekt der approximativen Gauss-Elimination beachtet werden, der durch das Zusammenspiel matrixabhängiger Prolongationen und Restriktionen mit einer hierarchischen Basis Zerlegung gegeben ist. Um alle diese Forderungen zu erfüllen, wird zunächst ausgehend von geometrischen Vergröberungen ein allgemeines Petrov--Galerkin Multiskalen-Konzept entwickelt, bei dem die Zerlegungen auf der Ansatz- und Testseite unterschiedlich sind. Es werden matrixabhängige Prolongationen, die von robusten Mehrgitter-Techniken her bekannt sind, verwendet, zusammen mit Wavelet-artigen und hierarchischen Multiskalen-Zerlegungen der Ansatz- und Testräume bezüglich des feinsten Gitters. Die Kernidee bei den vorgeschlagenen Verfahren ist, jeweils einen der Komplementräume auf der Ansatz- oder Testseite hierarchisch zu wählen, um zusammen mit einer problemabhängigen Vergröberung auf der anderen Seite physikalisch sinnvolle Grobgitter- diskretisierungen und gleichzeitig einen approximativen Eliminations- effekt zu erreichen. Die Komplementräume auf der entsprechend anderen Seite werden hingegen Wavelet-artig aufgespannt, was insbesondere zu einer Stabilisierung des Verfahrens bezüglich der Abhängigkeit von der Maschenweite der Diskretisierung führt. Mit den weiterhin entwickelten AMGlet-Zerlegungen, die auf rein algebraischen Prinzipien beruhen, gelingt es, geometrisch orientierte Tensorprodukt- Konstruktionen, die für separable Probleme erfolgreich sind, zu verlassen, um schwierige nichtseparable Aufgaben in unter Umständen kompliziert berandeten Gebieten behandeln zu können. Dies eröffnet darüberhinaus auch den Übergang von Modellproblemen hin zu praxisnahen Fragestellungen. Unterschiedliche numerische Beispiele zeigen, dass man durch die vorgeschlagenen Konstruktionen zu verallgemeinerten Hierarchische Basis Mehrgitter-Verfahren mit robusten Konvergenzeigenschaften gelangt