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Title The h-p version of the finite element method in three dimensions
Publication Date
Date Accessioned
Discipline/Department Mathematics
Degree Level doctor of philosophy (ph.d.)
University/Publisher University of Manitoba
Abstract 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.
Subjects/Keywords h-p version
Contributors Guo, Benqi (Mathematics) (supervisor); Lui, S.H. (Mathematics), Thulasiram, R. (Computer Science), Han, B. (Mathematics) (examiningcommittee)
Country of Publication ca
Record ID handle:1993/11753
Repository manitoba
Date Indexed 2019-05-20
Issued Date 2012-11-21 00:00:00
Note [note] October 2008;

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…1. INTRODUCTION 2 domains. After two-decades effort people realize very recently that the most appropriate mathematical framework for error analysis of the p and h-p version is the Jacobi-weighted Besov and Sobolev spaces. In a series of papers by…

…Guo and his collaborators [3, 4, 5, 6, 34], a new analysis of the p and h-p version was given in which the approximation theory of the FEM and BEM in two dimensions in this new mathematical framework was systematically developed. It…

…demonstrates that Jacobi-weighted Besov space is the most appropriate tool to obtain optimal upper and lower bounds when dealing with singular solutions on polygons. This framework has been generalized to the p-version and the h-p version of the BEM [32…

…33]. Thus the approximation theory for the p and h-p version of FEM and BEM in two dimensions has been established in the framework of the Jocobi-weighted Besov and Sobolev spaces. Although significant progresses for the p and h-p versions FEM in…

…one and two dimensions have been made in the past three decades, the approximation theory of the p and h-p versions of FEM in three dimensions is much less developed due to the complexity of three dimensional problems, and only a few results are…

…elements in three dimensions which realize global continuity of piecewise polynomial and remain the best approximation of local Jacobi projections. In despite of theoretical difficulties the computation and algorithms of the p and h-p FEM have made…

…possible to validate mathematical models and to verify the computational results. Since the p and h-p versions FEM in three dimensions provide higher accuracy and reduce significantly computational cost, they have been applied to various fields of…

…engineering and sciences such as mechanics, magnetoelectric, biology and material science [15, 37, 39, 43, 46], and have been implemented in new codes and enhanced in the existing codes. These codes with three dimensional p and h-p FEM capacity have…