In the first part, an efficient and reliable a posteriori error estimate is derived for solving three-dimensional static Maxwell's equations based on the lowest order edge elements of the first family. We propose an adaptive finite element method and establish convergence of the adaptive scheme in energy norm under a restriction on the initial mesh size. Any prescribed error tolerance is thus achieved in a finite number of steps. For discretization based on the lowest order edge elements of the second family, a similar adaptive method is designed which guarantees convergence without any initial mesh size restriction. The proofs rely mainly on error and oscillation reduction estimates as well as the Galerkin orthogonality of the edge element approximation. For time-dependent Maxwell's equations, we deduce an efficient and reliable a posteriori error estimate, upon which an adaptive finite element method is built. / In this thesis, we will address three typical problems with discontinuous coefficients in a general Lipschitz polyhedral domain, which are often encountered in numerical simulation of electromagnetism. / The second part deals with a saddle point problem arising from Maxwell's equations. We present an adaptive finite element method on the basis of the lowest order edge elements of the first family and prove its convergence. The main ingredients of the proof are a novel quasi-orthogonality, which replaces the usual Pythagoras relation, which fails in this case, all error reduction depending on an efficient and reliable a posteriori error estimate and an oscillation reduction. We show that this adaptive scheme is a contraction for the sum of some energy error plus the oscillation. Likewise, the above result is generalized to the discretization by the lowest order edge elements of the second family. / We introduce in the third part an adaptive finite element method for solving the eigenvalue problem of the Maxwell system based on an inverse iterative method. By modifying the exact inverse iteration algorithm involving an inner saddle point solver, we construct an adaptive inverse iteration finite element algorithm, which consists of an inexact inner adaptive procedure for a discrete mixed formulation in place of the original saddle point problem. An efficient and reliable a posteriori error estimate is obtained and the convergence of the inner adaptive method is proved. In addition, the important convergence property of the algorithm is studied, which ensures the errors between true solutions (eigenfunction and eigenvalue) and iterative ones to fall below any given tolerance within a finite number of iterations. / Xu, Yifeng. / "June 2007." / Adviser: Jun Zou. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0357. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 166-175). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_343980 |
| Date | January 2007 |
| Contributors | Xu, Yifeng, Chinese University of Hong Kong Graduate School. Division of Mathematics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, theses |
| Format | electronic resource, microform, microfiche, 1 online resource (ix, 175 p.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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