The largest and most difficult numerical approximation error to estimate is discretization error. Residual-based discretization error estimation methods are a category of error estimators that use an estimate of the source of discretization error and information about the specific application to estimate the discretization error using only one grid level. The higher-order terms are truncated from the discretized equations and are the local source of discretization error. The accuracy of the resulting discretization error estimate depends solely on the accuracy of the estimated truncation error. Residual-based methods require only one grid level compared to the more commonly used Richardson extrapolation which requires at least two. Reducing the required number of grid levels reduces computational expense and, since only one grid level is required, can be applied to unstructured grids where multiple quality grid levels are difficult to produce. The two residual-based discretization error estimators of interest are defect correction and error transport equations. The focus of this work is the development, improvement, and evaluation of various truncation error estimation methods considering the accuracy of the truncation error estimate and the resulting discretization error estimates. The minimum requirements for accurate truncation error estimation is specified along with proper treatment for several boundary conditions. The methods are evaluated using various Euler and Navier-Stokes applications. The discretization error estimates are compared to Richardson extrapolation. The most accurate truncation error estimation method was found to be the k-exact method where the fine grid with a correction factor was considerably reliable. The single grid methods including the k-exact require that the continuous operator be modified at the boundary to be consistent with the implemented boundary conditions. Defect correction showed to be more accurate for areas of larger discretization error; however, the cost was substantial (although cheaper than the primal problem) compared to the cost of solving the ETEs which was essential free due to the linearization. Both methods showed significantly more accurate estimates compared to Richardson extrapolation especially for smooth problems. Reduced accuracy was apparent with the presence of stronger shocks and some possible modifications to adapt to singularies are proposed for future work. / Ph. D.

Numerical solutions obtained through the use of Computational Fluid Dynamics (CFD) are subject to discretization error, which is locally generated by truncation error. The discretization error is extremely difficult to properly estimate and this in turn leads to uncertainty over the quality of the numerical solutions obtained via CFD methods and the engineering functionals computed using these solutions. Adjoint error estimation techniques specifically seek to estimate the error in functionals, but are dependent upon accurate truncation error estimates. This work examines the application of new, single-grid, truncation error estimation procedures to the problem of adjoint error estimation for both the quasi-1D and 2D Euler equations. The new truncation error estimation techniques are based on local reconstructions of the computed solutions and comparisons are made for the quasi-1D study in order to determine the most appropriate solution variables to reconstruct as well as the most appropriate reconstruction method. In addition, comparisons are made between the single-grid truncation error estimates and methods based on uniformally refining or coarsening the underlying numerical mesh on which the computed solutions are obtained. A method based on an refined grid error estimate is shown to work well for a non-isentropic flow for the quasi-1D Euler equations, but all truncation error estimations methods ultimately result in over prediction of functional discretization error in the presence of a shock in 2D. Alternatives to adjoint methods, which can only estimate the error in a single functional for each adjoint solution obtained, are examined for the 2D Euler equations. The defection correction method and error transport equations are capable of locally improving the entire computed solution, allowing for error estimates in multiple functionals. It is found that all three functional discretization error estimates perform similarly for the same truncation error estimate, although the defect correction method is the most costly from a computational viewpoint. Comparisons are made between truncation error and adjoint weighted truncation error based adaptive indicators. For the quasi-1D Euler equations it is found that both methods are competitive, however the truncation error based method is cheaper as a separate adjoint solve is avoided. For the 2D Euler equations, the truncation error estimates on the adapted meshes suffer due to a lack of smooth grid transformations which are used in reconstructing the computed solutions. In order to complete this work, a new CFD code incorporating a variety of best practices from the field of Computer Science is developed as well as a new method of performing code verification using the method of manufactured solutions which is significantly easier to implement than traditional manufactured solution techniques. / Ph. D.

Le but de cette étude est d'améliorer les performances, en termes d'espace mémoire et de temps de calcul, des simulations actuelles de l'Interaction mécanique Pastille-Gaine (IPG), phénomène complexe pouvant avoir lieu lors de fortes montées en puissance dans les réacteurs à eau sous pression. Parmi les méthodes de raffinement de maillage, méthodes permettant de simuler efficacement des singularités locales, une approche multi-grille locale a été choisie car elle présente l'intérêt de pouvoir utiliser le solveur en boîte noire tout en ayant un faible nombre de degrés de liberté à traiter par niveau. La méthode Local Defect Correction (LDC), adaptée à une discrétisation de type éléments finis, a tout d'abord été analysée et vérifiée en élasticité linéaire, sur des configurations issues de l'IPG, car son utilisation en mécanique des solides est peu répandue. Différentes stratégies concernant la mise en oeuvre pratique de l'algorithme multi-niveaux ont également été comparées. La combinaison de la méthode LDC et de l'estimateur d'erreur a posteriori de Zienkiewicz-Zhu, permettant d'automatiser la détection des zones à raffiner, a ensuite été testée. Les performances obtenues sur des cas bidimensionnels et tridimensionnels sont très satisfaisantes, l'algorithme proposé se montrant plus performant que des méthodes de raffinement h-adaptatives. Enfin, l'algorithme a été étendu à des problèmes mécaniques non linéaires. Les questions d'un raffinement espace/temps mais aussi de la transmission des conditions initiales lors du remaillage ont entre autres été abordées. Les premiers résultats obtenus sont encourageants et démontrent l'intérêt de la méthode LDC pour des calculs d'IPG. / The aim of this study is to improve the performances, in terms of memory space and computational time, of the current modelling of the Pellet-Cladding mechanical Interaction (PCI),complex phenomenon which may occurs during high power rises in pressurised water reactors. Among the mesh refinement methods - methods dedicated to efficiently treat local singularities - a local multi-grid approach was selected because it enables the use of a black-box solver while dealing few degrees of freedom at each level. The Local Defect Correction (LDC) method, well suited to a finite element discretisation, was first analysed and checked in linear elasticity, on configurations resulting from the PCI, since its use in solid mechanics is little widespread. Various strategies concerning the implementation of the multilevel algorithm were also compared. Coupling the LDC method with the Zienkiewicz-Zhu a posteriori error estimator in orderto automatically detect the zones to be refined, was then tested. Performances obtained on two-dimensional and three-dimensional cases are very satisfactory, since the algorithm proposed is more efficient than h-adaptive refinement methods. Lastly, the LDC algorithm was extended to nonlinear mechanics. Space/time refinement as well as transmission of the initial conditions during the remeshing step were looked at. The first results obtained are encouraging and show the interest of using the LDC method for PCI modelling.

Wir untersuchen ein Defektkorrekturverfahren, das ein einfaches Upwind-Differenzenverfahren erster Ordnung mit einem zentralen Differenzenverfahren kombiniert, für ein- und zweidimensionale singulär gestörte Konvektions-Diffusions-Probleme auf einer Klasse von Shishkin-Typ-Gittern. Im eindimensionalen Fall wird nachgewiesen, dass das Verfahren von (fast) zweiter Ordnung, gleichmäßig bezüglich des Diffusionsparameters $\epsilon$ konvergiert. Zur Konvergenzanalyse für das zweidimensionale Modellproblem werden verschiedene Techniken diskutiert. In einem Spezialfall kann auf einem stückweise uniformen Shishkin-Gitter die $\epsilon$-gleichmäßige Konvergenz des Verfahrens von fast zweiter Ordnung gezeigt werden. Ferner sind die bisher bekannten Stabilitätsaussagen und ihre Verwendung zur Konvergenzanalysis der betrachteten Differenzenverfahren sowie Methoden zur Analyse von Defektkorrekturverfahren zusammengestellt. Einige Bemerkungen zu Defektkorrekturverfahren und Finite-Elemente-Methoden schließen die Arbeit ab. Numerische Experimente untermauern die theoretischen Resultate. / We consider a defect correction method that combines a first-order upwinded difference scheme with a second-order central difference scheme for model singularly perturbed convection-diffusion problems in one and two dimensions on a class of Shishkin-Type meshes. In one dimension, the method is shown to be convergent uniformly in the diffusion parameter $\epsilon$ of second order in the discrete maximum norm. To analyze the two-dimensional case, we discuss several proof techniques for defect correction methods. For a special problem with constant coefficients on a piecewise uniform Shishkin-mesh we can show the second order convergence of the considered scheme, uniformly with respect to the diffusion parameter. Moreover the known stability properties and their impact on the convergence analysis of the considered differnce schemes are compiled. Some remarks on defect correction and finite elements conclude the theses. Numerical experiments support our theoretical results.

The development of numerical methods for the incompressible Navier-Stokes equations received much attention in the past 50 years. Finite element methods emerged given their robustness and reliability. In our work, we choose the P2-P1 finite element for space approximation which gives 2nd-order accuracy for velocity and 1st-order accuracy for pressure. Our research focuses on the development of several high-order semi-implicit time-stepping methods to compute unsteady flows. The methods investigated include backward difference formulae (SBDF) and defect correction strategy (DC). Using the defect correction strategy, we investigate two variants, the first one being based on high-order artificial compressibility and bootstrapping strategy proposed by Guermond and Minev (GM) and the other being a combination of GM methods with sequential regularization method (GM-SRM). Both GM and GM-SRM methods avoid solving saddle point problems as for SBDF and DC methods. This approach reduces the complexity of the linear systems at the expense that many smaller linear systems need to be solved. Next, we proposed several numerical improvements in terms of better approximations of the nonlinear advection term and high-order initialization for all methods. To further minimize the complexity of the resulting linear systems, we developed several new variants of grad-div splitting algorithms besides the one studied by Guermond and Minev. Splitting algorithm allows us to handle larger flow problems. We showed that our new methods are capable of reproducing flow characteristics (e.g., lift and drag parameters and Strouhal numbers) published in the literature for 2D lid-driven cavity and 2D flow around the cylinder. SBDF methods with grad-div stabilization terms are found to be very stable, accurate and efficient when computing flows with high Reynolds numbers. Lastly, we showcased the robustness of our methods to carry 3D computations.

Wir untersuchen ein Defektkorrekturverfahren, das ein einfaches Upwind-Differenzenverfahren erster Ordnung mit einem zentralen Differenzenverfahren kombiniert, für ein- und zweidimensionale singulär gestörte Konvektions-Diffusions-Probleme auf einer Klasse von Shishkin-Typ-Gittern. Im eindimensionalen Fall wird nachgewiesen, dass das Verfahren von (fast) zweiter Ordnung, gleichmäßig bezüglich des Diffusionsparameters $\epsilon$ konvergiert. Zur Konvergenzanalyse für das zweidimensionale Modellproblem werden verschiedene Techniken diskutiert. In einem Spezialfall kann auf einem stückweise uniformen Shishkin-Gitter die $\epsilon$-gleichmäßige Konvergenz des Verfahrens von fast zweiter Ordnung gezeigt werden. Ferner sind die bisher bekannten Stabilitätsaussagen und ihre Verwendung zur Konvergenzanalysis der betrachteten Differenzenverfahren sowie Methoden zur Analyse von Defektkorrekturverfahren zusammengestellt. Einige Bemerkungen zu Defektkorrekturverfahren und Finite-Elemente-Methoden schließen die Arbeit ab. Numerische Experimente untermauern die theoretischen Resultate. / We consider a defect correction method that combines a first-order upwinded difference scheme with a second-order central difference scheme for model singularly perturbed convection-diffusion problems in one and two dimensions on a class of Shishkin-Type meshes. In one dimension, the method is shown to be convergent uniformly in the diffusion parameter $\epsilon$ of second order in the discrete maximum norm. To analyze the two-dimensional case, we discuss several proof techniques for defect correction methods. For a special problem with constant coefficients on a piecewise uniform Shishkin-mesh we can show the second order convergence of the considered scheme, uniformly with respect to the diffusion parameter. Moreover the known stability properties and their impact on the convergence analysis of the considered differnce schemes are compiled. Some remarks on defect correction and finite elements conclude the theses. Numerical experiments support our theoretical results.

Le but de cette étude est d'améliorer les performances, en termes d'espace mémoire et de temps de calcul, des simulations actuelles de l'Interaction mécanique Pastille-Gaine (IPG), phénomène complexe pouvant avoir lieu lors de fortes montées en puissance dans les réacteurs à eau sous pression. Parmi les méthodes de raffinement de maillage, méthodes permettant de simuler efficacement des singularités locales, une approche multi-grille locale a été choisie car elle présente l'intérêt de pouvoir utiliser le solveur en boîte noire tout en ayant un faible nombre de degrés de liberté à traiter par niveau. La méthode Local Defect Correction (LDC), adaptée à une discrétisation de type éléments finis, a tout d'abord été analysée et vérifiée en élasticité linéaire, sur des configurations issues de l'IPG, car son utilisation en mécanique des solides est peu répandue. Différentes stratégies concernant la mise en oeuvre pratique de l'algorithme multi-niveaux ont également été comparées. La combinaison de la méthode LDC et de l'estimateur d'erreur a posteriori de Zienkiewicz-Zhu, permettant d'automatiser la détection des zones à raffiner, a ensuite été testée. Les performances obtenues sur des cas bidimensionnels et tridimensionnels sont très satisfaisantes, l'algorithme proposé se montrant plus performant que des méthodes de raffinement h-adaptatives. Enfin, l'algorithme a été étendu à des problèmes mécaniques non linéaires. Les questions d'un raffinement espace/temps mais aussi de la transmission des conditions initiales lors du remaillage ont entre autres été abordées. Les premiers résultats obtenus sont encourageants et démontrent l'intérêt de la méthode LDC pour des calculs d'IPG.