In our project we discuss a finite element time-relaxation method for high Reynolds number flows. The key idea consists of using local projections on polynomials defined on macro element of each pair of two elements sharing a face. We give the formulation for the scalar convection–diffusion equation and a numerical illustration.

For more than 100 years, the Navier-Stokes equations and various linearizations have been used as a model to study fluid dynamics. Recently, attention has been directed toward studying the nonnormality of linearized problems and developing convergent numerical schemes for simulation of these sytems. Numerical schemes for optimal control problems often require additional properties that may not be necessary for simulation; these properties can be critical when studying nonnormal problems. This research is concerned with approximating infinite dimensional optimal control problems with nonnormal system operators. We examine three different finite element methods for a specific convection-diffusion equation and prove convergence of the infinitesimal generators. Additionally, for two of these schemes, we prove convergence of the associated feedback gains. We apply these three schemes to control problems and compare the performance of all three methods. / Ph. D.

A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.

Dans le contexte du stockage des déchets radioactifs en milieu poreux, nous considérons l’équation de convection-diffusion instationnaire et sa discrétisation par des méthodes numériques. La discontinuité des paramètres physiques et la variabilité des échelles d’espace et de temps conduisent à utiliser des discrétisations différentes en temps et en espace dans différentes régions du domaine. Nous choisissons dans cette thèse le schéma volumes finis en dualité discrète (DDFV) et le schéma de Galerkin Discontinu en temps couplés à une méthode de décomposition de domaine de Schwarz de type relaxation d’ondes optimisées (OSWR), ce qui permet de traiter des maillages espace-temps non conformes. La principale difficulté réside dans l’obtention d’une discrétisation amont du flux convectif qui reste locale à un sous-domaine et telle que le schéma monodomaine soit équivalent au schéma multidomaine. Ces difficultés sont appréhendées d’abord en une dimension d’espace où différentes discrétisations sont étudiées. Le schéma retenu introduit une inconnue hybride sur les interfaces entre cellules. L’idée du décentrage amont par rapport à cette inconnue hybride est reprise en dimension deux d’espace, et adaptée au schéma DDFV. Le caractère bien posé de ce schéma et d’un schéma multidomaine équivalent est montré. Ce dernier est résolu par un algorithme OSWR dont la convergence est prouvée. Les paramètres optimisés des conditions de Robin sont obtenus par l'étude du taux de convergence continu ou discret. Différents cas-tests, dont l’un est inspiré du stockage des déchets nucléaires, illustrent ces résultats. / In the context of nuclear waste repositories, we consider the numerical discretization of the non stationary convection diffusion equation. Discontinuous physical parameters and heterogeneous space and time scales lead us to use different space and time discretizations in different parts of the domain. In this work, we choose the discrete duality finite volume (DDFV) scheme and the discontinuous Galerkin scheme in time, coupled by an optimized Scwharz waveform relaxation (OSWR) domain decomposition method, because this allows the use of non-conforming space-time meshes. The main difficulty lies in finding an upwind discretization of the convective flux which remains local to a sub-domain and such that the multidomain scheme is equivalent to the monodomain one. These difficulties are first dealt with in the one-dimensional context, where different discretizations are studied. The chosen scheme introduces a hybrid unknown on the cell interfaces. The idea of upwinding with respect to this hybrid unknown is extended to the DDFV scheme in the two-dimensional setting. The well-posedness of the scheme and of an equivalent multidomain scheme is shown. The latter is solved by an OSWR algorithm, the convergence of which is proved. The optimized parameters in the Robin transmission conditions are obtained by studying the continuous or discrete convergence rates. Several test-cases, one of which inspired by nuclear waste repositories, illustrate these results.

We study the behavior of numerical stabilization schemes in the context of linear quadratic regulator (LQR) control problems for convection diffusion equations. The motivation for this effort comes from the observation that when linearization is applied to fluid flow control problems the resulting equations have the form of a convection diffusion equation. This effort is focused on the specific problem of computing the feedback functional gains that are the kernels of the feedback operators defined by solutions of operator Riccati equations. We develop a stabilization scheme based on the Galerkin Least Squares (GLS) method and compare this scheme to the standard Galerkin finite element method. We use cubic B-splines in order to keep the higher order terms that occur in GLS formulation. We conduct a careful numerical investigation into the convergence and accuracy of the functional gains computed using stabilization. We also conduct numerical studies of the role that the stabilization parameter plays in this convergence. Overall, we discovered that stabilization produces much better approximations to the functional gains on coarse meshes than the unstabilized method and that adjustments in the stabilization parameter greatly effects the accuracy and convergence rates. We discovered that the optimal stabilization parameter for simulation and steady state analysis is not necessarily optimal for solving the Riccati equation that defines the functional gains. Finally, we suggest that the stabilized GLS method might provide good initial values for iterative schemes on coarse meshes. / Ph. D.

A singularly perturbed convection-diffusion problem in two and three space dimensions is discretized using the streamline upwind Petrov Galerkin (SUPG) variant of the finite element method. The dominant convection frequently gives rise to solutions with layers; hence anisotropic finite elements can be applied advantageously. The main focus is on a posteriori energy norm error estimation that is robust in the perturbation parameter and with respect to the mesh anisotropy. A residual error estimator and a local problem error estimator are proposed and investigated. The analysis reveals that the upper error bound depends on the alignment of the anisotropies of the mesh and of the solution. Hence reliable error estimation is possible for suitable anisotropic meshes. The lower error bound depends on the problem data via a local mesh Peclet number. Thus efficient error estimation is achieved for small mesh Peclet numbers. Altogether, error estimation approaches for isotropic meshes are successfully extended to anisotropic elements. Several numerical experiments support the analysis.

Thesis submitted in fulfilment of the requirements for the degree Doctor technologiae (engineering: chemical) In the faculty of engineering at the cape peninsula university of technology / Since the first uses of hollow-fiber membrane bioreactors (MBR’s) to immobilize whole cells were reported in the early 1970’s, this technology has been used in as wide ranging applications as enzyme production to bone tissue engineering. The potential of these devices in industrial applications is often diminished by the large diffusional resistances of the membranes. Currently, there are no analytical studies on the performance of the MBR which account for both convective and diffusive transport. The purpose of this study was to quantify the efficiency of a biocatalytic membrane reactor used for the production of enzymes. This was done by developing exact solutions of the concentration and velocity profiles in the different regions of the membrane bioreactor (MBR). The emphasis of this study was on the influence of radial convective flows, which have generally been neglected in previous analytical studies. The efficiency of the MBR was measured by means of the effectiveness factor. An analytical model for substrate concentration profiles in the lumen of the MBR was developed. The model was based on the solution of the Navier-Stokes equations and Darcy’s law for velocity profiles, and the convective-diffusion equation for the solute concentration profiles. The model allowed for the evaluation of the influence of both hydrodynamic and mass transfer operating parameters on the performance of the MBR. These parameters include the fraction retentate, the transmembrane pressure, the membrane hydraulic permeability, the Reynolds number, the axial and radial Peclet numbers, and the dimensions of the MBR. The significant findings on the hydrodynamic studies were on the influence of the fraction retentate. In the dead-end mode it was found that there was increased radial convective flow, and hence more solute contact with the enzymes/biofilm immobilised on the surface of the membrane. The improved solute-biofilm contact however was only limited to the entrance half of the MBR. In the closed shell mode there was uniform distribution of solute, however, radial convective flows were significantly reduced. The developed model therefore allowed for the evaluation of an optimum fraction retentate value, where both the distribution of solutes and radial convective flows could be maximised.

The sequence-dependent curvature and flexibility of DNA is critical for many biochemically important processes. However, few experimental methods are available for directly probing these properties at the base-pair level. One promising way to predict these properties as a function of sequence is to model DNA with a set of base-pair parameters that describe the local stacking of the different possible base-pair step combinations. In this dissertation research, we develop and study a computational model for predicting the diffusion coefficients of short, relatively rigid DNA fragments from the sequence and the base-pair parameters. We focus on diffusion coefficients because various experimental methods have been developed to measure them. Moreover, these coefficients can also be computed numerically from the Stokes equations based on the three-dimensional shape of the macromolecule. By comparing the predicted diffusion coefficients with experimental measurements, we can potentially obtain refined estimates of various base-pair parameters for DNA. Our proposed model consists of three sub-models. First, we consider the geometric model of DNA, which is sequence-dependent and controlled by a set of base-pair parameters. We introduce a set of new base-pair parameters, which are convenient for computation and lead to a precise geometric interpretation. Initial estimates for these parameters are adapted from crystallographic data. With these parameters, we can translate a DNA sequence into a curved tube of uniform radius with hemispherical end caps, which approximates the effective hydrated surface of the molecule. Second, we consider the solvent model, which captures the hydrodynamic properties of DNA based on its geometric shape. We show that the Stokes equations are the leading-order, time-averaged equations in the particle body frame assuming that the Reynolds number is small. We propose an efficient boundary element method with a priori error estimates for the solution of the exterior Stokes equations. Lastly, we consider the diffusion model, which relates our computed results from the solvent model to relevant measurements from various experimental methods. We study the diffusive dynamics of rigid particles of arbitrary shape which often involves arbitrary cross- and self-coupling between translational and rotational degrees of freedom. We use scaling and perturbation analysis to characterize the dynamics at time scales relevant to different classic experimental methods and identify the corresponding diffusion coefficients. In the end, we give rigorous proofs for the convergence of our numerical scheme and show numerical evidence to support the validity of our proposed models by making comparisons with experimental data. / text

Title: Adaptive choice of parameters in stabilization methods for convection- diffusion equations Author: Bc. Petr Lukáš (e-mail: luk.p@post.cz) Department: Department of Numerical Mathematics Supervisor: Doc. Mgr. Petr Knobloch, Dr. (e-mail: knobloch@karlin.mff.cuni.cz) Abstract: The aim of the work is to propose suitable approaches for adap- tive choice of parameters in stabilization methods for convection-difusion equations discretized by the finite element method. We introduce the L-SR1 method, compare it with other nonlinear methods of minimizing functions with large number of variables, and introduce and compare the adaptive methods based on minimizing of the error indicator. Keywords: Adaptive choice of parameters, finite element method, stabiliza- tion methods, convection-diffusion equation, L-SR1 method, error indicator

La surveillance des rejets gazeux des installations nucléaires dans l'environnement et de contrôle des dispositifs d'épuration reposent sur des mesures régulières de concentrations des contaminants en sortie de cheminées et dans les réseaux de ventilation. La répartition de la concentration peut être hétérogène au niveau du point de mesure si la distance d'établissement du mélange est insuffisante. La question se pose sur l'évaluation du positionnement des points de piquage et sur l'erreur commise par rapport à la concentration homogène en cas de non-respect de cette distance. Cette étude définit cette longueur dite de « bon mélange » à partir d'expériences menées en laboratoire. Le banc dimensionné pour ces essais a permis de reproduire des écoulements dans des conduits longs circulaire et rectangulaire, comprenant chacun un coude. Une technique de mesure optique a été développée, calibrée puis utilisée pour mesurer la distribution de la concentration d'un traceur injecté dans l'écoulement. Les résultats expérimentaux en conduit cylindrique ont validé un modèle analytique basé sur l'équation de convection-diffusion d'un traceur, et ont permis de proposer des modèles de longueur de bon mélange et de représentativité de points de prélèvement. Dans le conduit à section rectangulaire, les mesures acquises constituent une première base de données sur l'évolution de l'homogénéisation d'un traceur, dans la perspective de simulations numériques explorant des conditions plus réalistes des mesures in situ. / Monitoring of gaseous releases from nuclear installations in the environment and air cleaning efficiency measurement are based on regular measurements of concentrations of contaminants in outlet chimneys and ventilation systems. The concentration distribution may be heterogeneous at the measuring point if the distance setting of the mixing is not sufficient. The question is about the set up of the measuring point in duct and the error compared to the homogeneous concentration in case of non-compliance with this distance. This study defines the so-called "well mixing length" from laboratory experiments. The bench designed for these tests allowed to reproduce flows in long circular and rectangular ducts, each including a bend. An optical measurement technique has been developed, calibrated and used to measure the concentration distribution of a tracer injected in the flow. The experimental results in cylindrical duct have validated an analytical model based on the convection-diffusion equation of a tracer, and allowed to propose models of good mixing length and representativeness of sampling points. In rectangular duct, the acquired measures constitute a first database on the evolution of the homogenization of a tracer, in the perspective of numerical simulations exploring more realistic conditions for measurements in situ.