This work examines the Method of Nearby Problems as a way to generate analytical exact solutions to problems governed by partial differential equations (PDEs). The method involves generating a numerical solution to the original problem of interest, curve fitting the solution, and generating source terms by operating the governing PDEs upon the curve fit. Adding these source terms to the right-hand-side of the governing PDEs defines the nearby problem. In addition to its use for generating exact solutions the MNP can be extended for use as an error estimator. The nearby problem can be solved numerically on the same grid as the original problem. The nearby problem discretization error is calculated as the difference between its numerical solution and exact solution (curve fit). This is an estimate of the discretization error in the original problem of interest. The accuracy of the curve fits is quite important to this work. A method of curve fitting that takes local least squares fits and combines them together with weighting functions is used. This results in a piecewise fit with continuity at interface boundaries. A one-dimensional Burgers' equation case shows this to be a better approach then global curve fits. Six two-dimensional cases are investigated including solutions to the time-varying Burgers' equation and to the 2D steady Euler equations. The results show that the Method of Nearby Problems can be used to create realistic, analytical exact solutions to problems governed by PDEs. The resulting discretization error estimates are also shown to be reasonable for several cases examined. / Master of Science

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.

Computational fluid dynamics (CFD) simulations can provide tremendous insight into complex physical processes and are often faster and more cost-effective to execute than experiments. However, each CFD result inherently contains numerical errors that can significantly degrade the accuracy of a simulation. Discretization error is typically the largest contributor to the overall numerical error in a given simulation. Discretization error can be very difficult to estimate since the generation, transport, and diffusion of these errors is a highly nonlinear function of the computational grid and discretization scheme. As CFD is increasingly used in engineering design and analysis, it is imperative that CFD practitioners be able to accurately quantify discretization errors to minimize risk and improve the performance of engineering systems. In this work, improvements are made to the accuracy and efficiency of existing error estimation techniques. Discretization error is estimated by deriving and solving an error transport equation (ETE) for the local discretization error everywhere in the computational domain. Truncation error is shown to act as the local source for discretization error in numerical solutions. An equivalence between adjoint methods and ETE methods for functional error estimation is presented. This adjoint/ETE equivalence is exploited to efficiently obtain error estimates for multiple output functionals and to extend the higher-order properties of adjoint methods to ETE methods. Higher-order discretization error estimates are obtained when truncation error estimates are sufficiently accurate. Truncation error estimates are demonstrated to deteriorate on grids with a non-smooth variation in grid metrics (e.g., unstructured grids) regardless of how smooth the underlying exact solution may be. The loss of accuracy is shown to stem from noise in the discrete solution on the order of discretization error. When using conventional least-squares reconstruction techniques, this noise is exactly captured and introduces a lower-order error into the truncation error estimate. A novel reconstruction method based on polyharmonic smoothing splines is developed to smoothly reconstruct the discrete solution and improve the accuracy of error estimates. Furthermore, a method for iteratively improving discretization error estimates is devised. Efficiency and robustness considerations are discussed. Results are presented for several inviscid and viscous flow problems. To facilitate the study of discretization error estimation, a new, higher-order finite-volume solver is developed. A detailed description of the code base is provided along with a discussion of best practices for CFD code design. / Ph. D. / Computational fluid dynamics (CFD) is a branch of computational physics at the intersection of fluid mechanics and scientific computing in which the governing equations of fluid motion, such as the Euler and Navier-Stokes equations, are solved numerically on a computer. CFD is utilized in numerous fields including biomedical engineering, meteorology, oceanography, and aerospace engineering. CFD simulations can provide tremendous insight into physical processes and are often preferred over experiments because they can be performed more quickly, are typically more cost-effective, and can provide data in regions where it may be difficult to measure. While CFD can be an extremely powerful tool, CFD simulations are inherently subject to numerical errors. These errors, which are generated when the governing equations of fluid motion are solved on a computer, can have a significant impact on the accuracy of a CFD solution. If numerical errors are not accurately quantified, ill-informed decision-making can lead to poor system performance, increased risk of injury, or even system failure. In this work, research efforts are focused on numerical error estimation for the finite -volume method, arguably the most widely used numerical algorithm for solving CFD problems. The error estimation techniques provided herein target discretization error, the largest contributor to the overall numerical error in a given simulation. Discretization error can be very difficult to estimate since these errors are generated, convected, and diffused by the same physical processes embedded in the governing equations. In this work, improvements are made to the accuracy and efficiency of existing discretization error estimation techniques. Results are presented for several inviscid and viscous flow problems. To facilitate the study of these error estimators, a new, higher-order finite -volume solver is developed. A detailed description of the code base is provided along with a discussion of best practices for CFD code design.

L’objectif principal de la thèse est de répondre à des questions liées aux étapes clé d’un processus de l’adaptation de modèles de plaques. Comme l’adaptativité dépend des estimateurs d’erreurs fiables, une part importante du rapport est dédiée au développement des méthodes numériques pour les estimateurs d’erreurs aussi bien dues à la discrétisation qu’au choix du modèle. Une comparaison des estimateurs d’erreurs de discrétisation d’un point de vue pratique est présentée. Une attention particulière est prêtée a la méthode de résiduels équilibrés (en anglais, "equilibrated residual method"), laquelle est potentiellement applicable aux estimations des deux types d’erreurs, de discrétisation et de modèle.Il faut souligner que, contrairement aux estimateurs d’erreurs de discrétisation, les estimateurs d’erreur de modèle sont plus difficiles à élaborer. Le concept de l’adaptativité de modèles pour les plaques est implémenté sur la base de la méthode de résiduels équilibrés et de la famille hiérarchique des éléments finis de plaques. Les éléments finis dérivés dans le cadre de la thèse, comprennent aussi bien les éléments de plaques minces et que les éléments de plaques épaisses. Ces derniers sont formulés en s’appuyant sur une théorie nouvelle de plaque, intégrant aussi les effets d’étirement le long de l’épaisseur. Les erreurs de modèle sont estimées via des calculs élément par élément. Les erreurs de discrétisation et de modèle sont estimées d’une manière indépendante, ce qui rend l’approche très robuste et facile à utiliser. Les méthodes développées sont appliquées sur plusieurs exemples numériques. Les travaux réalisés dans le cadre de la thèse représentent plusieurs contributions qui visent l’objectif final de la modélisation adaptative, ou une procédure complètement automatique permettrait de faire un choix optimal du modèle de plaques pour chaque élément de la structure. / The primary goal of the thesis is to provide some answers to the questions related to the key steps in the process of adaptive modeling of plates. Since the adaptivity depends on reliable error estimates, a large part of the thesis is related to the derivation of computational procedures for discretization error estimates as well as model error estimates. A practical comparison of some of the established discretization error estimates is made. Special attention is paid to what is called equilibrated residuum method, which has a potential to be used both for discretization error and model error estimates. It should be emphasized that the model error estimates are quite hard to obtain, in contrast to the discretization error estimates. The concept of model adaptivity for plates is in this work implemented on the basis of equilibrated residuum method and hierarchic family of plate finite element models.The finite elements used in the thesis range from thin plate finite elements to thick plate finite elements. The latter are based on a newly derived higher order plate theory, which includes through the thickness stretching. The model error is estimated by local element-wise computations. As all the finite elements, representing the chosen plate mathematical models, are re-derived in order to share the same interpolation bases, the difference between the local computations can be attributed mainly to the model error. This choice of finite elements enables effective computation of the model error estimate and improves the robustness of the adaptive modeling. Thus the discretization error can be computed by an independent procedure.Many numerical examples are provided as an illustration of performance of the derived plate elements, the derived discretization error procedures and the derived modeling error procedure. Since the basic goal of modeling in engineering is to produce an effective model, which will produce the most accurate results with the minimum input data, the need for the adaptive modeling will always be present. In this view, the present work is a contribution to the final goal of the finite element modeling of plate structures: a fully automatic adaptive procedure for the construction of an optimal computational model (an optimal finite element mesh and an optimal choice of a plate model for each element of the mesh) for a given plate structure.

Title: Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor of the master thesis: Zdeněk Strakoš Abstract: After introduction of the model problem and its properties we describe the Conjugate Gradient Method (CG). We present the estimates of the energy norm of the error and a heuristic for the adaptive refinement of the estimate. The difference in the local behaviour of the discretization and the algebraic error is illustrated by numerical experiments using the given model problem. A posteriori estimates for the discretization and the total error that take into account the inexact solution of the algebraic system are then discussed. In order to get a useful perspective, we briefly recall the multigrid method. Then the Cascadic Conjugate Gradient Method of Deuflhard (CCG) is presented. Using the estimates for the error presented in the preceding parts of the thesis, the new stopping criteria for CCG are proposed. The CCG method with the new stopping criteria is then tested. Keywords: numerical PDE, discretization error, algebraic error, error es- timates, locality of the error, adaptivity