Theses and Dissertations at Montana State University (MSU)
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Item Sinc domain decomposition methods for elliptic problems(Montana State University - Bozeman, College of Letters & Science, 1994) Lybeck, Nancy JeanItem A fully Galerkin method for parabolic problems(Montana State University - Bozeman, College of Letters & Science, 1989) Lewis, David LamarItem The maximum deflection of a blast loaded cantilever beam by the modified Galerkin method(Montana State University - Bozeman, College of Engineering, 1969) Koszuta, Daniel MichaelItem Extensions to the development of the Sinc-Galerkin method for parabolic problems(Montana State University - Bozeman, College of Letters & Science, 1990) Doyle, Randy RossItem A sinc-collocation method for Burgers' equation(Montana State University - Bozeman, College of Letters & Science, 1995) Carlson, Timothy ScottItem Judging the relative qualities and merits of Galerkin's approximate solutions to a dynamically loaded beam system(Montana State University - Bozeman, College of Engineering, 1970) Hanson, Thomas MichaelGalerkin's method is applied to a beam structure that is forced to nonlinear behavior by a dynamic load. Nonlinearities in the system include a nonlinear stress-strain relation and consideration of geometry changes due to large deflections. Several trial deflection shapes are assumed as approximate solutions of the problem and these trial shapes are combined in various manners in an effort to produce a better quality solution. All resulting solutions are studied in the light of three criteria that are postulated in an attempt to define the relative merits and qualities of approximate solutions. It is concluded that although the criteria are good guidelines to finding reasonable solutions, they are not strict in defining a good quality solution.Item Sinc-Galerkin solution of second-order hyperbolic problems in multiple space dimensions(Montana State University - Bozeman, College of Letters & Science, 1987) McArthur, Kelly MarieItem An Alternating-Direction Sinc-Galerkin method for elliptic problems on finite and infinite domains(Montana State University - Bozeman, College of Letters & Science, 2009) Alonso, Nicomedes, III; Chairperson, Graduate Committee: Kenneth L. BowersAlternating-Direction Implicit (ADI) schemes are a class of very efficient algorithms for the numerical solution of differential equations. Sinc-Galerkin schemes employ a sinc basis to produce exponentially accurate approximate solutions to differential equations even in the presence of singularities. In this dissertation we begin with a broad overview of sinc methods for problems posed on both finite and infinite, one- and two-dimensional domains. We then present a variety of finite difference methods that lead to the introduction of a new Alternating-Direction Sinc-Galerkin scheme based on the classic ADI scheme for a linear matrix system. We note that when a Sinc-Galerkin method is used to solve a Poisson equation, the resulting matrix system is a Sylvester equation. We discuss ADI model problems in general and then prove that when a symmetric Sinc-Galerkin method is employed, the resulting Sylvester equation can be classified as an ADI model problem. Finally, we derive our Alternating-Direction Sinc-Galerkin (ADSG) method to solve this resulting Sylvester equation, specifying the use of a constant iteration parameter to avoid costly eigen-value computations. We end by applying ADSG to a variety of problems, comparing its performance to the standard technique that uses the Kronecker product, the Kronecker sum, and the concatenation operator.Item Comparision of continuous and discontinuous Galerkin finite element methods for parabolic partial differential equations with implicit time stepping(Montana State University - Bozeman, College of Engineering, 2012) Vo, Garret Dan; Chairperson, Graduate Committee: Jeffrey HeysA number of different discretization techniques and algorithms have been developed for approximating the solution of parabolic partial differential equations. A standard approach, especially for applications that involve complex geometries, is the classic continuous Galerkin finite element method. This approach has a strong theoretical foundation and has been widely and successfully applied to this category of differential equations. One challenging sub-category of problems, however, are equations that include an advection term that is large relative to the second-order, diffusive term. For these advection dominated problems, the continuous Galerkin finite element method can become unstable and yield highly inaccurate results. An alternative to the continuous Galerkin finite element method is the discontinuous Galerkin finite element method, and, through the use of a numerical flux term used in deriving the weak form, the discontinuous approach has the potential to be much more stable in highly advective problems. However, the discontinuous Galerkin finite element method also has significantly more degrees-of-freedom due to the replication of nodes along element edges and vertices. The work presented here compares the computational cost, stability, and accuracy (when possible) of continuous and discontinuous Galerkin finite element methods for four different test problems, including the advection-diffusion equation, viscous Burgers' equation, and the Turing pattern formation equation system. The comparison is performed using as much shared code as possible between the two algorithms and direct, iterative, and multilevel linear solvers. The results show that, for implicit time stepping, the continuous Galerkin finite element method is typically 5-20 times less computationally expensive than the discontinuous Galerkin finite element method using the same finite element mesh and element order. However, the discontinuous Galerkin finite element method is significantly more stable than the continuous Galerkin finite element method for advection dominated problems and is able to obtain accurate approximate solutions for cases where the classic, un-stabilized continuous Galerkin finite element method fails.