Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme

This is the second part of our error analysis of the stabilized Lagrange-Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi-Pitk\"aranta's stabilization method for the conforming linear elements, which leads to an efficient computation with a small number of degrees of freedom especially in three space dimensions. In this paper, Part II, we apply a semi-implicit time discretization which yields the linear scheme. We concentrate on the diffusive viscoelastic model, i.e. in the constitutive equation for time evolution of the conformation tensor a diffusive effect is included. Under mild stability conditions we obtain error estimates with the optimal convergence order for the velocity, pressure and conformation tensor in two and three space dimensions. The theoretical convergence orders are confirmed by numerical experiments.Comment: See arXiv:1603.01339 for Part I: a nonlinear schem

Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part I: A nonlinear scheme

We present a nonlinear stabilized Lagrange-Galerkin scheme for the Oseen-type Peterlin viscoelastic model. Our scheme is a combination of the method of characteristics and Brezzi-Pitk\"aranta's stabilization method for the conforming linear elements, which yields an efficient computation with a small number of degrees of freedom. We prove error estimates with the optimal convergence order without any relation between the time increment and the mesh size. The result is valid for both the diffusive and non-diffusive models for the conformation tensor in two space dimensions. We introduce an additional term that yields a suitable structural property and allows us to obtain required energy estimate. The theoretical convergence orders are confirmed by numerical experiments. In a forthcoming paper, Part II, a linear scheme is proposed and the corresponding error estimates are proved in two and three space dimensions for the diffusive model.Comment: See arXiv:1603.01074 for Part II: a linear schem

Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations

Error estimates with optimal convergence orders are proved for a stabilized Lagrange-Galerkin scheme for the Navier-Stokes equations. The scheme is a combination of Lagrange-Galerkin method and Brezzi-Pitkaranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The theoretical convergence orders are recognized numerically by two- and three-dimensional computations

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

Generalized local projection stabilized nonconforming finite element methods for Darcy equations

An a priori analysis for a generalized local projection stabilized finite element solution of the Darcy equations is presented in this paper. A first-order nonconforming P nc 1 finite element space is used to approximate the velocity, whereas the pressure is approximated using two different finite elements, namely piecewise constant P0 and piecewise linear nonconforming P nc 1 elements. The considered finite element pairs, P nc 1 /P0 and P nc 1 /P nc 1 , are inconsistent and incompatibility, respectively, for the Darcy problem. The stabilized discrete bilinear form satisfies an inf-sup condition with a generalized local projection norm. Moreover, a priori error estimates are established for both finite element pairs. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples

Mathematical Aspects of Computational Fluid Dynamics

[no abstract available