CUHK electronic theses & dissertations collection

In this thesis, we consider four different stochastic partial differential equations. Firstly, we study stochastic Helmholtz equation driven by an additive white noise, in a bounded convex domain with smooth boundary of Rd (d = 2, 3). And then with the help of the perfectly matched layers technique, we also consider the stochastic scattering problem of Helmholtz type. The second part of this thesis is to investigate the time harmonic case for stochastic Maxwell's equations driven by an color noise in a simple medium, and then we expand the results to the stochastic Maxwell's equations in case of dispersive media in Rd (d = 2, 3). Thirdly, we study stochastic parabolic partial differential equation driven by space-time color noise, where the domain O is a bounded domain in R2 with boundary ∂O of class C2+alpha for 0 < alpha < 1/2. In the last part, we discuss the stochastic wave equation (SWE) driven by nonlinear noise in 1D case, where the noise 626x6t W(x, t) is the space-time mixed second-order derivative of the Brownian sheet.Many physical and engineering phenomena are modeled by partial differential equations which often contain some levels of uncertainty. The advantage of modeling using so-called stochastic partial differential equations (SPDEs) is that SPDEs are able to more fully capture interesting phenomena; it also means that the corresponding numerical analysis of the model will require new tools to model the systems, produce the solutions, and analyze the information stored within the solutions.One of the goals of this thesis is to derive error estimates for numerical solutions of the above four kinds SPDEs. The difficulty in the error analysis in finite element methods and general numerical approximations for a SPDE is the lack of regularity of its solution. To overcome such a difficulty, we follow the approach of [4] by first discretizing the noise and then applying standard finite element methods and discontinuous Galerkin methods to the stochastic Helmholtz equation and Maxwell equations with discretized noise; standard finite element method to the stochastic parabolic equation with discretized color noise; Galerkin method to the stochastic wave equation with discretized white noise, and we obtain error estimates are comparable to the error estimates of finite difference schemes.We shall focus on some SPDEs where randomness only affects the right-hand sides of the equations. To solve the four types of SPDEs using, for example, the Monte Carlo method, one needs many solvers for the deterministic problem with multiple right-hand sides. We present several efficient deterministic solvers such as flexible CG method and block flexible GMRES method, which are absolutely essential in computing statistical quantities.Zhang, Kai.Adviser: Zou Jun.Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3552.Thesis (Ph.D.)--Chinese University of Hong Kong, 2008.Includes bibliographical references (leaves 144-155).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