Browsing by Subject "least-squares"
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Item A discontinuous least-squares spatial discretization for the sn equations(2009-05-15) Zhu, LeiIn this thesis, we develop and test a fundamentally new linear-discontinuous least-squares (LDLS) method for spatial discretization of the one-dimensional (1-D) discrete-ordinates (SN) equations. This new scheme is based upon a least-squares method with a discontinuous trial space. We implement our new method, as well as the lineardiscontinuous Galerkin (LDG) method and the lumped linear-discontinuous Galerkin (LLDG) method. The implementation is in FORTRAN. We run a series of numerical tests to study the robustness, L2 accuracy, and the thick diffusion limit performance of the new LDLS method. By robustness we mean the resistance to negativities and rapid damping of oscillations. Computational results indicate that the LDLS method yields a uniform second-order error. It is more robust than the LDG method and more accurate than the LLDG method. However, it fails to preserve the thick diffusion limit. Consequently, it is viable for neutronics but not for radiative transfer since radiative transfer problems can be highly diffusive.Item Least-squares methods for computational electromagnetics(Texas A&M University, 2004-11-15) Kolev, Tzanio ValentinovThe modeling of electromagnetic phenomena described by the Maxwell's equations is of critical importance in many practical applications. The numerical simulation of these equations is challenging and much more involved than initially believed. Consequently, many discretization techniques, most of them quite complicated, have been proposed. In this dissertation, we present and analyze a new methodology for approximation of the time-harmonic Maxwell's equations. It is an extension of the negative-norm least-squares finite element approach which has been applied successfully to a variety of other problems. The main advantages of our method are that it uses simple, piecewise polynomial, finite element spaces, while giving quasi-optimal approximation, even for solutions with low regularity (such as the ones found in practical applications). The numerical solution can be efficiently computed using standard and well-known tools, such as iterative methods and eigensolvers for symmetric and positive definite systems (e.g. PCG and LOBPCG) and reconditioners for second-order problems (e.g. Multigrid). Additionally, approximation of varying polynomial degrees is allowed and spurious eigenmodes are provably avoided. We consider the following problems related to the Maxwell's equations in the frequency domain: the magnetostatic problem, the electrostatic problem, the eigenvalue problem and the full time-harmonic system. For each of these problems, we present a natural (very) weak variational formulation assuming minimal regularity of the solution. In each case, we prove error estimates for the approximation with two different discrete least-squares methods. We also show how to deal with problems posed on domains that are multiply connected or have multiple boundary components. Besides the theoretical analysis of the methods, the dissertation provides various numerical results in two and three dimensions that illustrate and support the theory.Item Least-squares variational principles and the finite element method: theory, formulations, and models for solid and fluid mechanics(Texas A&M University, 2004-09-30) Pontaza, Juan PabloWe consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solidand fluidmechanics.For manyof these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model.Most notably, the use of least-squares principles leads to a variational unconstrained minimization problem where stability conditions such as inf-sup conditions (typically arising in mixed methods using weak form Galerkin finite element formulations) never arise. In addition, the least-squares based finite elementmodelalways yields a discrete system ofequations witha symmetric positive definite coeffcientmatrix.These attributes, amongst manyothers highlightedand detailed in this work, allow the developmentofrobust andeffcient finite elementmodels for problems of practical importance. The research documented herein encompasses least-squares based formulations for incompressible and compressible viscous fluid flow, the bending of thin and thick plates, and for the analysis of shear-deformable shell structures.