Efficient mapped spectral methods for unbounded and exterior domains
Many problems of fundamental and practical importance in science and engineering are naturally set in unbounded and exterior domains. Incompressible fluid dynamics, acoustic and electromagnetic wave scattering and computational optional pricing are typical examples of this type. Such problems pre...
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| 格式: | Theses and Dissertations |
| 語言: | English |
| 出版: |
2008
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| 主題: | |
| 在線閱讀: | http://hdl.handle.net/10356/14517 |
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| 總結: | Many problems of fundamental and practical importance in science and engineering are
naturally set in unbounded and exterior domains. Incompressible fluid dynamics, acoustic
and electromagnetic wave scattering and computational optional pricing are typical examples
of this type. Such problems present a great challenge to computational scientists and
numerical analysts partially due to the unboundedness of the computational domains ruling
out the direct use of many numerical methods available for bounded domains. In the
past decades, considerable progress has been made in developing numerical methods for
unbounded/exterior domains. The viable methods include (1) domain truncation supplemented
with artificial boundary conditions and coupled with finite difference, finite elements
and spectral methods, and (2) direct methods using orthogonal systems on unbounded
domains. This research falls into the second category and aims to introduce a general
framework for analyzing and implementing spectral methods using mapping techniques for
unbounded and exterior domains. More precisely, a general type of mappings is applied
to a general family of orthogonal systems in finite interval, which generate a variety of
orthogonal basis functions for spectral methods for unbounded and exterior domains. The
approximation properties of the new bases in Sobolev spaces are analyzed in a unform fashion.
Specific mappings of exponential, logarithmic and algebraic types are studied under
this general setting. The framework leads to more concise analysis, and in most of cases,
optimal results. The effect of scaling fact is also featured in the approximation results.
These results provide quantitative criteria for the choice of mappings and parameters for
partial differential equations in unbounded/exterior domains. The general mapped spectral
methods are applied to some PDEs in unbounded and exterior domains. The performance
of some popular mappings are compared and assessed under the general framework. |
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