Augsburg, Univ., Diss., 2008.

University-Gesamthochsch., Diss.--Paderborn, 1995.

anglicky In the presented work, we study the existence and uniqueness of solutions to the generalized Stokes problem. We, further, focus on the higher differentiability and the Hölder continuity of solutions to the generalized Stokes and generalized Navier-Stokes system. We reach the full regularity in an arbitrary dimension for a linear case, while in a nonlinear case we work only in dimensions d = 2, 3. In dimension d = 2 we are able to proof the full regularity of solution, in dimension d = 3 we obtain only a partial regularity. All main results are introduced in the first section. 1

The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.

La notion de schémas gradients, conçue pour les équations elliptiques et paraboliques, linéaires et non-linéaires a l'avantage de fournir des résultats de convergence et d'estimations d'erreur valables pour de nombreuses familles de méthodes numériques (éléments finis conformes et non-conformes, éléments finis mixtes, différences finies ...). Vérifier un ensemble restreint de propriétés suffit pour prouver qu'une méthode numérique donnée rentre dans le cadre de travail des schémas gradients et donc qu'elle sera convergente sur les différents problèmes traités. L'étude du problème de Stefan, celle du problème de Stokes incompressible, ainsi que celle des équations de Navier-Stokes incompressibles sont présentées dans cette thèse, chacune présentant un théorème de convergence établi à l'aide des schémas gradients. Pour Stokes et Navier-Stokes, nous donnerons une preuve de convergence pour les cas stationnaires et transitoires en modifiant certaines hypothèses ce qui aura comme effet de trouver des résultats de convergence différents. Finalement, nous présentons également quatre méthodes (Taylor-Hood, Crouzeix-Raviart, Marker-and-Cell, Hybrid Mixed Mimetic) pour ces deux problèmes et nous vérifions qu'elles rentrent bien dans le cadre des schémas gradients / The notion of gradient schemes, designed for linear and nonlinear elliptic and parabolic problems has the benefit of providing common convergence and error estimates results, which hold for a wide variety of numerical methods (finite element methods, nonconforming and mixed finite element methods, hybrid and mixed mimetic finite difference methods ...). Checking a minimal set of properties for a given numerical method suffices to prove that it belongs to the gradient schemes framework, and therefore that it is convergent on the different problems studied here. The study of the Stefan problem, the incompressible Stokes one and also the incompressible Navier-Stokes equations are presented in this thesis, where each one gets a convergence theorem set up with the gradient schemes framework. For Stokes and Navier-Stokes, we both provide the proof for the steady and the transient case dealing with some variational hypotheses which bring different convergence results. Finally, we also present four methods (Taylor-Hood, Crouzeix-Raviart, Marker-and-Cell, Hybrid Mixed Mimetic) for these two problems and we check that they enter in the gradient schemes framework

In this thesis, the seminal work of Stokes concerning the ow of a Navier-Stokesuid due to a suddenly accelerated or oscillating plate and the ow due to torsionaloscillations of an innitely long rod in a Navier-Stokes uid is extended to a uid withpressure dependent viscosity. The viscosity of many uids varies signicantly withpressure, a fact recognized by Stokes; and Barus, in fact, conducted experiments thatshowed that the variation of the viscosity with pressure was exponential. Given sucha tremendous variation, we study how this change in viscosity aects the nature of thesolution to these problems. We nd that the velocity eld, and hence the structureof the vorticity and the shear stress at the walls for uids with pressure dependentviscosity, are markedly dierent from those for the classical Navier-Stokes uid.

In this thesis, the seminal work of Stokes concerning the ow of a Navier-Stokesuid due to a suddenly accelerated or oscillating plate and the ow due to torsionaloscillations of an innitely long rod in a Navier-Stokes uid is extended to a uid withpressure dependent viscosity. The viscosity of many uids varies signicantly withpressure, a fact recognized by Stokes; and Barus, in fact, conducted experiments thatshowed that the variation of the viscosity with pressure was exponential. Given sucha tremendous variation, we study how this change in viscosity aects the nature of thesolution to these problems. We nd that the velocity eld, and hence the structureof the vorticity and the shear stress at the walls for uids with pressure dependentviscosity, are markedly dierent from those for the classical Navier-Stokes uid.

The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and nonconforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.

In the context of gradient-based numerical optimisation, the adjoint method is an e cient way of computing the gradient of the cost function at a computational cost independent of the number of design parameters, which makes it a captivating option for industrial CFD applications involving costly primal solves. The method is however a ected by instabilities, some of which are inherited from the primal solver, notably if the latter does not fully converge. The present work is an attempt at curbing primal solver limitations with the goal of indirectly alleviating adjoint robustness issues. To that end, a novel discretisation scheme for the steady-state incompressible Navier- Stokes problem is proposed: Mixed Hybrid Finite Volumes (MHFV). The scheme draws inspiration from the family of Mimetic Finite Di erences and Mixed Virtual Elements strategies, rid of some limitations and numerical artefacts typical of classical Finite Volumes which may hinder convergence properties. Derivation of MHFV operators is illustrated and each scheme is validated via manufactured solutions: rst for pure anisotropic di usion problems, then convection-di usion-reaction and nally Navier-Stokes. Traditional and novel Navier-Stokes solution algorithms are also investigated, adapted to MHFV and compared in terms of performance. The attention is then turned to the discrete adjoint Navier-Stokes system, which is assembled in an automated way following the principles of Equational Di erentiation, i.e. the di erentiation of the primal discrete equations themselves rather than the algorithm used to solve them. Practical/computational aspects of the assembly are discussed, then the adjoint gradient is validated and a few solution algorithms for the MHFV adjoint Navier-Stokes are proposed and tested. Finally, two examples of full shape optimisation procedures on internal ow test cases (S-bend and U-bend) are reported.

Cette thèse comprend deux parties principales:La première partie est une étude du problème d'élasticité linéaire en temps par une méthode de Galerkin discontinue (SIPG). Dans cette partie, nous avons toutd'abord obtenu un estimateur a posteriori pour la formulation semi-discrète. En utilisant une technique de reconstruction et des résultats montrés dans le cas stationnaire, on a établi un estimateur a posteriori d'erreur pour le problème d'onde élastique dépendant du temps. Afin de calculer l'estimateur d'erreur lié au cas stationnaire, nous avons présenté deux méthodes, l'une utilisant la technique de la dualité ce qui nous a donné un calcul d'erreur en norme L^2 et l'autre en calculant l'erreur en norme énergie. Pour la discrétisation en temps l'équation, nous utilisons un schéma numérique d'Euler. En utilisant une technique et de reconstruction spatio-temporelle, on propose un nouvel estimateur a posteriori.La deuxième partie a pour but l'établissement d'un développement asymptotiquepour la solution de problème résolvant Stokes avec une petite perturbation dudomaine. Dans ce travail, nous avons appliqué la théorie du potentiel. On a écrit la solution du problème non perturbé et du problème perturbé sous forme d'opérateurs intégraux. En calculant la différence, et en utilisant des propriétés liées aux noyaux des opérateurs on a établi un développement asymptotique de la solution. / This thesis contains two main parts:The first part concerning the discontinuous Galerkin method for the timedependentlinear elasticity problem. In this part, we have derived the a posteriorierror bounds for semi-discrete and fully discrete formulation, by makinguse of the SE reconstruction technique which allows to estimate the errorbound for time-dependent problem through the error estimation of the ascociatedstationary elasticity problem. Then to derive the error bound for thestationary problem, we have presented two methods to obtain two different aposteriori bounds, by L2 duality technique and via energy norm. For fully discretescheme, we make use of the backward-Euler scheme and an appropiatespace-time reconstruction which has the zero-mean value in time.The second part concerning the derivation of an asymptotic expansionfor the solution of Stokes resolvent problem with a small perturbation of thedomain. In this work, we have applied the potential theory, boundary integralequation method and geometric properties of perturbed boundary. Thederivation is rigorous, and this method allows to derive high-order terms inasymptotic expansion. Also, it can be used for many other boundary valueproblems, whenever a suitable potential theory is available.