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.

Master of Science