Finite Elemente gleicher Ordnung von hydrostatischen Ströungsproblemen

Subject of this thesis is the issue of equal-order finite element discretization of hydrostatic flow problems. These flow problems typically arise in geophysical fluid dynamics on large scales and in flat domains. This small aspect ratio between the depth and the horizontal extents of the considered domain allows to efficiently reduce the complexity of the incompressible three dimensional Navier-Stokes equations, which form the basis of geophysical flows. In the resulting set of equations, the vertical momentum equation is replaced by the hydrostatic balance, which thus decouples the vertical pressure variations from the dynamic system, and the dynamically relevant pressure becomes two dimensional. Moreover, the vertical velocity component can be explicitely determined by the horizontal velocity components. Concomitant with this reduction is the replacement of the divergence constraint by a suitably modified version of it. As in the classical framework, it is known that these hydrostatic flow problems also show a saddle point structure, and there is a similar uncertainty concerning existence and uniqueness of solutions as is apparent for the classical case. Although the variational framework has been intensively treated, the issue of the discretization, in particular the finite element discretization of hydrostatic problems has hardly been considered yet. The present work dedicates to this topic. We indicate the tight relation between a finite element discretized hydrostatic flow problem and its two dimensional counterpart with respect to inf-sup stability. Moreover, we elaborate stabilization techniques in order to result to inf-sup stable schemes and to suitably treat the case of dominant advection. For each of these cases we can draw on classical stabilization schemes. For the isotropic hydrostatic Stokes problem we thus derive and examine residual-based as well as symmetric stabilization schemes. In the appropriate Oseen case we restrict to symmetric stabilization schemes. Beside the isotropic case, we also consider hydrostatic problems on vertical anisotropic meshes, i.e. although the mesh may be anisotropic, the surface mesh still shows isotropic structure. Therefore we derive an interpolation operator, which has suitable projection and stability properties in three dimensions. An appropriate operator for the two dimensional case for bilinear finite element spaces has been developed in Braack06. In this vertical anisotropic context we restrict to symmetric stabilizat on schemes for both problems, the hydrostatic Stokes and the hydrostatic Oseen problem. Further, we also examine the hydrostatic Stokes problem on meshes with anisotropy occurring also in the surface mesh. This may be necessary in regions with strong flows in one horizontal direction, e.g. in the Bering strait or along coastlines. In a following chapter we shortly discuss on the time discretization approach, particularly on the issue of pressure correction schemes. These schemes are discussed already in a couple of works for classical flow problems. But a proper analysis is still missing. Finally, after considering algorithmic aspects, which also includes the topic of parallelization, we numerically validate our theoretical results and numerically illustrate apparent physical phenomena occurring in ocean circulation regimes.Die vorliegende Arbeit widmet sich der Thematik der Diskretisierung von hydrostatischen Strömungsproblemen mittels Finiter Elemente gleicher Ordnung. Hydrostatische Strömungsprobleme treten typischerweise im Bereich der geophysikalischen Fluiddynamik auf grossen Skalen und in flachen Gebieten auf. Mathematische Grundlage bilden die inkompressiblen dreidimensionalen (3D) Navier-Stokes Gleichungen. Das kleine Aspektverhältnis zwischen der Gebietstiefe und der horizontalen Ausdehnung des Gebietes erlaubt es, die Komplexität der inkompressiblen 3D Navier-Stokes Gleichungen merkbar zu reduzieren. Anwendung der sogenannten hydrostatischen Approximation, welches das kleine Aspektverhältnis ausnutzt, führt dazu, dass die vertikale Gleichung der Impulserhaltung durch die hydrostatische Balance ersetzt wird. Dadurch wird der dynamisch relevante Druck zweidimensional (2D) und die vertikale Geschwindigkeit bestimmt sich direkt aus den horizontalen. Einhergehend mit dieser Reduktion ist eine Modifikation der Bedingung der Divergenzfreiheit. Das resultierende hydrostatische Strömungsproblem weist bekanntermaßen eine Sattelpunktstruktur auf, ähnlich dem klassischen Problem. Desweiteren herrscht auch im hydrostatischen Kontext eine ähnliche Unsicherheit bezüglich Existenz und Eindeutigkeit von Lösungen vor, wie sie auch in der klassischen Navier-Stokes-Thematik anzutreffen ist. Obwohl hydrostatische Probleme im variationellen Rahmen intensiv untersucht worden sind und werden, ist das Feld der Diskretisierung dieser Probleme, insbesondere die Finite-Elemente-Diskretisierung, größtenteils unbearbeitet. Die vorliegende Arbeit widmet sich dieser Thematik. Wir zeigen die enge Beziehung auf, die bezüglich der Inf-sup-Stabilität zwischen dem diskreten hydrostatischen Strömungsproblem und seinem 2D Pendant existiert. Desweiteren erarbeiten wir Stabilisierungsverfahren, um Inf-sup-Stabilität zu erlangen und den Fall der dominanten Advektion adäquat zu behandeln. Hierbei können wir auf klassische Stabilisierungsverfahren zurückgreifen. Neben dem isotropen Fall betrachten wir hydrostatische Probleme auf anisotropen Gittern. Für die Analyse entwickeln wir einen Interpolationsoperator, der passende Projektions- und Stabilitätseigenschaften in 3D besitzt. Ein entsprechender Operator für den 2D Fall für bilineare Finite Elemente wurde in Braack06 entwickelt. Für die Stabilisierung beschränken wir uns auf symmetrische Verfahren. Die Druckstabilisierung bleibt aufgrund der Dimension des Drucks auf vertikal anisotropen Gitter, d.h. obwohl Gitteranisotropie auftreten kann ist das Oberflächengitter isotrop, isotrop. Im Fall auftretender Gitteranisotropie auch im Horizontalen greifen wir auf anisotrope Druckstabilisierung zurück. Desweitern diskutieren wir kurz die Thematik der Zeitdiskretisierung. Insbesondere gehen wir auf Druckkorrektur-Verfahren ein. Diese Verfahren wurden bereits für klassische Strömungsprobleme diskutiert. Jedoch fehlt bislang eine Analyse dieser Thematik im hydrostatischen Kontext. Anschließend betrachten wir algorithmische Aspekte und gehen dabei auch auf die Thematik der Parallelisierung ein. Wir schließen die Arbeit mit einer numerischen Validierung der theoretischen Ergebnisse ab und illustrieren einige Phänomene der Ozeanzirkulation

A Computational Framework for Axisymmetric Linear Elasticity and Parallel Iterative Solvers for Two-Phase Navier–Stokes

This dissertation explores ways to improve the computational efficiency of linear elasticity and the variable density/viscosity Navier--Stokes equations. While the approaches explored for these two problems are much different in nature, the end goal is the same - to reduce the computational effort required to form reliable numerical approximations.\\ The first topic considered is the axisymmetric linear elasticity problem. While the linear elasticity problem has been studied extensively in the finite-element literature, to the author\u27s knowledge, this is the first study of the elasticity problem in an axisymmetric setting. Indeed, the axisymmetric nature of the problem means that a change of variables to cylindrical coordinates reduces a three-dimensional problem into a decoupled one-dimensional and two-dimensional problem. The change of variables to cylindrical coordinates, however, affects the functional form of the divergence operator and the definition of the inner products. To develop a computational framework for the linear elasticity problem in this context, a new projection operator is defined that is tailored to the cylindrical form of the divergence and inner products. Using this framework, a stable finite-element quadruple is derived for $k=1,2$. These computational rates are then validated with a few computational examples.\\ The second topic addressed in this work is the development of a new Schur complement approach for preconditioning the two-phase Navier--Stokes equations. Considerable research effort has been invested in the development of Schur complement preconditioning techniques for the Navier--Stokes equations, with the pressure-convection diffusion (PCD) operator and the least-squares commutator being among the most popular. Furthermore, more recently researchers have begun examining preconditioning strategies for variable density / viscosity Stokes and Navier--Stokes equations. This work contributes to recent work that has extended the PCD Schur complement approach for single phase flow to the variable phase case. Specifically, this work studies the effectiveness of a new two-phase PCD operator when applied to dynamic two-phase simulations that use the two-phase Navier--Stokes equations. To demonstrate the new two-phase PCD operators effectiveness, results are presented for standard benchmark problems, as well as parallel scaling results are presented for large-scale dynamic simulations for three-dimensional problems

Domain decomposition methods for the coupling of surface and groundwater flows

The purpose of this thesis is to investigate, from both the mathematical and numerical viewpoint, the coupling of surface and porous media flows, with particular concern on environmental applications. Domain decomposition methods are applied to set up effective iterative algorithms for the numerical solution of the global problem. To this aim, we reformulate the coupled problem in terms of an interface (Steklov-Poincaré) equation and we investigate the properties of the Steklov-Poincaré operators in order to characterize optimal preconditioners that, at the discrete level, yield convergence in a number of iterations independent of the mesh size h. We consider a new approach to the classical Robin-Robin method and we reinterpret it as an alternating direction iterative algorithm. This allows us to characterize robust preconditioners for the linear Stokes/Darcy problem which improve the behaviour of the classical Dirichlet- Neumann and Neumann-Neumann ones. Several numerical tests are presented to assess the convergence properties of the proposed algorithms. Finally, the nonlinear Navier-Stokes/Darcy coupling is investigated and a general nonlinear domain decomposition strategy is proposed for the solution of the interface problem, extending the usual Newton or fixed-point based algorithms