In the framework of the Jacobi-weighted Besov and Sobolev spaces, we analyze the approximation to singular and smooth functions. We construct stable and compatible polynomial extensions from triangular and square faces to prisms, hexahedrons and pyramids, and introduce quasi Jacobi projection operators on individual elements, which is a combination of the Jacobi projection and the interpolation at vertices and on sides of elements. Applying these results we establish the convergence of the h-p version of the finite element method with quasi uniform meshes in three dimensions for elliptic problems with smooth solutions or singular solutions on polyhedral domains in three dimensions. The rate of convergence interms of h and p is proved to be the best.

In the framework of the Jacobi-weighted Besov and Sobolev spaces, we analyze the approximation to singular and smooth functions. We construct stable and compatible polynomial extensions from triangular and square faces to prisms, hexahedrons and pyramids, and introduce quasi Jacobi projection operators on individual elements, which is a combination of the Jacobi projection and the interpolation at vertices and on sides of elements. Applying these results we establish the convergence of the h-p version of the finite element method with quasi uniform meshes in three dimensions for elliptic problems with smooth solutions or singular solutions on polyhedral domains in three dimensions. The rate of convergence interms of h and p is proved to be the best.

In this thesis, least-squares based finite element models (LSFEM) for the Poisson equation and Navier-Stokes equation are presented. The least-squares method is simple, general and reliable. Least-squares formulations offer several computational and theoretical advantages. The resulting coefficient matrix is symmetric and positive-definite. Using these formulations, the choice of approximating space is not subject to any compatibility condition. The Poisson equation is cast as a set of first order equations involving gradient of the primary variable as auxiliary variables for the mixed least-square finite element model. Equal order C0 continuous approximation functions is used for primary and auxiliary variables. Least-squares principle was directly applied to develop another model which requires C1continous approximation functions for the primary variable. Each developed model is compared with the conventional model to verify its performance. Penalty based least-squares formulation was implemented to develop a finite element for the Navier Stokes equations. The continuity equation is treated as a constraint on the velocity field and the constraint is enforced using the penalty method. Velocity gradients are introduced as auxiliary variables to get the first order equivalent system. Both the primary and auxiliary variables are interpolated using equal order C0 continuous, p-version approximation functions. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.

In this paper, we consider domain decomposition preconditioners for a system of linear algebraic equations arising from the <i>p</i>-version of the fem. We propose several multi-level preconditioners for the Dirichlet problems in the sub-domains in two and three dimensions. It is proved that the condition number of the preconditioned system is bounded by a constant independent of the polynomial degree. The proof uses interpretations of the <i>p</i>-version element stiffness matrix and mass matrix on [-1,1] as <i>h</i>-version stiffness matrix and weighted mass matrix. The analysis requires wavelet methods.

La plupart des problèmes de la physique et de la mécanique conduisent à des équations aux dérivées partielles. Les nombreuses méthodes qui existent déjà sont de degré relativement bas. Dans cette thèse, nous proposons une méthode de très haut degré. Notre idée est d'augmenter l'ordre des fonctions d'interpolation via une technique de perturbation afin d'éviter ou de réduire les difficultés engendrées par les opérations très coûteuses comme les intégrations. En dimension 1, la technique proposée est proche de la P-version des éléments finis. Au niveau élémentaire, on approxime la solution par une série entière d'ordre p. Dans le cas d'une équation linéaire d'ordre 2, cette résolution locale permet de construire un élément de degré élevé, avec deux degrés de liberté par élément. Pour les problèmes nonlinéaires, une linéarisation du problème par la méthode de Newton s'impose. Des tests portant sur des équations linéaires et nonlinéaires ont permis de valider la méthode et de montrer que la technique a une convergence similaire à la p-version des éléments finis. En dimension 2, le problème se discrétise grâce à une réorganisation des polynômes en polynômes homogènes de degré k. Après une définition de variables dites principales et secondaires associé à un balayage vertical du domaine, le problème devient une suite de problème 1D. Une technique de collocation permet de prendre en compte les conditions aux limites et les conditions de raccord et de déterminer la solution du problème. La collocation couplée avec la technique des moindres carrés a permis de d'améliorer les premiers résultats et a ainsi rendu plus robuste la technique de perturbation / Most problems of physics and mechanics lead to partial differential equations. The many methods that exist are relatively low degree. In this thesis, we propose a method of very high degree. Our idea is to increase the order of interpolation function via a perturbation technique to avoid or reduce the difficulties caused by the high cost operations such as integrations. In dimension 1, the proposed technique is close to the P-version finite elements. At a basic level, approximates the solution by a power series of order p. In the case of a linear equation of order 2, the local resolution can build an element of degree, with two degrees of freedom per element. For nonlinear problems, a linearization of the problem by Newton's method is needed. Tests involving linear and nonlinear equations were used to validate the method and show that the technique has a similar convergence in the p-version finite elements. In dimension 2, the problem is discretized through reorganizing polynomials in homogeneous polynomials of degree k. After a definition of variables called principal and secondary combined with a vertical scanning field, the problem becomes a series of 1D problem. A collocation technique allows to take into account the boundary conditions and coupling conditions and determine the solution of the problem. The collocation technique coupled with the least-squares enabled to improve the initial results and has made more robust the perturbation technique

In this paper, we consider domain decomposition preconditioners for a system of linear algebraic equations arising from the <i>p</i>-version of the fem. We propose several multi-level preconditioners for the Dirichlet problems in the sub-domains in two and three dimensions. It is proved that the condition number of the preconditioned system is bounded by a constant independent of the polynomial degree. The proof uses interpretations of the <i>p</i>-version element stiffness matrix and mass matrix on [-1,1] as <i>h</i>-version stiffness matrix and weighted mass matrix. The analysis requires wavelet methods.

Automation of finite element analysis based on a fully adaptive <i>p</i>-refinement procedure can reduce the magnitude of discretization error to the desired accuracy with minimum computational cost and computer resources. This study aims to 1) develop an efficient <i>p</i>-refinement procedure with a non-uniform <i>p</i> analysis capability for solving 2-D and 3-D elastostatic mechanics problems, and 2) introduce a stress error estimate. An element-by-element algorithm that decides the appropriate order for each element, where element orders can range from 1 to 8, is described. Global and element data structures that manage the complex data generated during the refinement process are introduced. These data structures are designed to match the concept of object-oriented programming where data and functions are managed and organized simultaneously. The stress error indicator introduced is found to be more reliable and to converge faster than the error indicator measured in an energy norm called the residual method. The use of the stress error indicator results in approximately 20% fewer degrees of freedom than the residual method. Agreement of the calculated stress error values and the stress error indicator values confirms the convergence of final stresses to the analyst. The error order of the stress error estimate is postulated to be one order higher than the error order of the error estimate using the residual method. The mapping of a curved boundary element in the working coordinate system from a square-shape element in the natural coordinate system results in a significant improvement in the accuracy of stress results. Numerical examples demonstrate that refinement using non-uniform <i>p</i> analysis is superior to uniform <i>p</i> analysis in the convergence rates of output stresses or related terms. Non-uniform <i>p</i> analysis uses approximately 50% to 80% less computational time than uniform <i>p</i> analysis in solving the selected stress concentration and stress intensity problems. More importantly, the non-uniform <i>p</i> refinement procedure scales the number of equations down by 1/2 to 3/4. Therefore, a small scale computer can be used to solve equation systems generated using high order <i>p</i>-elements. In the calculation of the stress intensity factor of a semi-elliptical surface crack in a finite-thickness plate, non-uniform <i>p</i> analysis used fewer degrees of freedom than a conventional <i>h</i>-type element analysis found in the literature. / Ph. D.