Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require $\mathcal{O}(p^{2d})$ storage and $\mathcal{O}(p^{3d})$ computational work, where $p$ is the degree of basis polynomials used, and $d$ is the spatial dimension. Our SVD-based tensor-product preconditioner requires $\mathcal{O}(p^{d+1})$ storage, $\mathcal{O}(p^{d+1})$ work in two spatial dimensions, and $\mathcal{O}(p^{d+2})$ work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in $p$ per degree of freedom in 2D, and reduce the computational complexity from $\mathcal{O}(p^9)$ to $\mathcal{O}(p^5)$ in 3D. Numerical results are shown in 2D and 3D for the advection and Euler equations, using polynomials of degree up to $p=15$. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees $p$.Comment: 40 pages, 15 figure

Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

Algebraic Multigrid for Meshfree Methods

This thesis deals with the development of a new Algebraic Multigrid method (AMG) for the solution of linear systems arising from Generalized Finite Difference Methods (GFDM). In particular, we consider the Finite Pointset Method, which is based on GFDM. Being a meshfree method, FPM does not rely on a mesh and can therefore deal with moving geometries and free surfaces is a natural way and it does not require the generation of a mesh before the actual simulation. In industrial use cases the size of the linear systems often becomes large, which means that classical linear solvers often become the bottleneck in terms of simulation run time, because their convergence rate depends on the discretization size. Multigrid methods have proven to be very efficient linear solvers in the domain of mesh-based methods. Their convergence is independent of the discretization size, yielding a run time that only scales linearly with the problem size. AMG methods are a natural candidate for the solution of the linear systems arising in the FPM, as this thesis will show. They need to be tuned to the specific characteristics of GFDM, though. The AMG methods that are developed in this thesis achieve a speed-up of up to 33x compared to the classical linear solvers and therefore allow much more accurate simulations in the future.Diese Dissertation beschäftigt sich mit der Entwicklung einer neuen Algebraischen Mehrgittermethode für die Lösung linearer Gleichungssysteme aus Generalisierten Finite Differenzen Methoden. Im Speziellen betrachten wir die sogenannte Finite Pointset Method, eine gitterfreie Lagrange Methode, welche auf Generalisierten Finite Differenzen Methoden basiert. Die Finite Pointset Method wurde insbesondere für Simulationen von Vorgängen mit freien Oberflächen und bewegten Geometrien entwickelt, bei denen der gitterfreie Charakter der Methode besonders große Vorteile liefert: An den freien Oberflächen und nahe der Geometrie muss zu keinem Zeitpunkt – auch nicht zu Beginn der Simulation – ein Gitter erstellt oder angepasst werden. Dies ist ein großer Vorteil gegenüber klassischen gitterbasierten Methoden. Wie in gitterbasierten Methoden entstehen auch in der Finite Pointset Method und anderen Generalisierten Finite Differenzen Methoden große, dünn besetze lineare Gleichungssysteme. Das Lösen dieser Gleichungssysteme wird bei fein aufgelösten Simulationen, wie sie in der Industrie oft nötig sind, schnell zum zeitlichen Flaschenhals der Gesamtsimulation. Ohne eine geeignete Methode zur Lösung dieser Gleichungssysteme dauern Simulationen oft sehr lange oder sind praktisch nicht durchführbar. Auch kann es vorkommen, dass klassische Lösungsverfahren divergieren und die Simulation damit unmöglich wird. Im Kontext von gitterbasierten Methoden sind Mehrgittermethoden ein etabliertes Werkzeug, um die entstehenden linearen Gleichungssysteme effizient und robust zu lösen. Besonders hervorzuheben ist dabei die lineare Skalierbarkeit dieser Methoden in der Größe der Matrix. Damit eignen sie sich besonders für fein aufgelöste Simulationen. Algebraische Mehrgittermethoden sind natürliche Kandidaten für die Lösung der Gleichungssysteme aus Generalisierten Finite Differenzen Methoden, wie diese Dissertation zeigen wird. Außerdem entwickeln wir eine neue Algebraische Mehrgittermethode, die auf den Einsatz in der Finite Pointset Method zugeschnitten ist und die Besonderheiten dieser Methode beachtet. Dazu zählen die Eigenschaften der einzelnen Matrizen, die wir ebenfalls analysieren werden, und auch die Veränderung der Matrizen über mehrere Zeitschritte hinweg, die im Vergleich mit gitterbasierten Verfahren eine größere Schwierigkeit darstellt. Wir evaluieren unsere neue Methode anhand von akademischen und realen Beispielen, sowohl mit nur einem Prozess als auch mit mehreren (MPI-)Prozessen. Die hier neu entwickelte Algebraische Mehrgittermethode ist um ein Vielfaches schneller als klassische Verfahren zur Lösung linearer Gleichungssysteme und erlaubt damit neue, genauere Simulationen mit gitterfreien Methoden

Parallel finite element modeling of the hydrodynamics in agitated tanks

Mixing in the transition flow regime -- Technology to mix in transition flow regime -- Methods to characterize mixing hydrodynamics -- Challenges to numerically model transition flow regime in agited tanks -- Transition flow regime in agitated tanks -- Parallel computing -- Numerical modeling of the agitators motion -- Overall methodological approach -- Computational resources -- Program development strategy -- Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers -- Parallel numerical model -- Parallel implementation -- Three-dimensional benchmark cases -- A parallel finite element sliding mesh technique for the Navier-Stokes equation -- Numerical method -- Parallel implementation -- Numerical examples -- Parallel performance -- Finite element modeling of the laminar and transition flow of the Superblend dual shaft coaxial mixer on parallel computers -- Superblend coaxial mixer configuration -- Numerical model -- Hydrodynamics in Superblend coaxial mixer -- Mixing -- Mixing efficiency -- Parallel finite element solver -- Parallel sliding mesh technique -- Simulation of the hydrodynamics of a stirred tank in the transition regime -- Recommendations for future research -- Parallel algorithms -- Simulation of agited and the transition flow regime

Solution strategies for nonlinear conservation laws

Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy. Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton's method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed. However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium. By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems. Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation

A stabilized edge-based implicit incompressible flow formulation

An edge-based implementation of an implicit, monolithic, finite element (FE) scheme for the solution of the incompressible Navier–Stokes (NS) equations is presented. The original element formulation is based on the pressure stability properties of an implicit second-order in time fractional step (FS) method, which is conditionally stable. The final monolithic scheme preserves the second-order accuracy of the FS method, and is unconditionally stable. Furthermore, it can be demonstrated that the final pressure stabilizing term is practically the same fourth-order pressure term added by some authors (but following different arguments) to obtain high order accurate results, and that the final discretized convective terms are formally a second-order discretization of the respective continuous one. The development of the edge implementation is supported by two criteria: the properties of the element based one, which has already been extensively tested and for which convergence and stability analysis has already been presented, and on the enforcement of global conservation and symmetry at the discrete level. A monotonicity preserving term which decreases the discretization order in sharp gradient regions to avoid localized oscillations (overshoots and undershoots), is formulated and tested. Some numerical examples and experimental comparisons are presente