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2012

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Mixed Finite Element Approximation Of The Vector Laplacian With Dirichlet Boundary Conditions, Douglas N. Arnold, Richard S. Falk, Jay Gopalakrishnan Jan 2012

Mixed Finite Element Approximation Of The Vector Laplacian With Dirichlet Boundary Conditions, Douglas N. Arnold, Richard S. Falk, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed …

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Wavenumber Explicit Analysis Of A Dpg Method For The Multidimensional Helmholtz Equation, Leszek Demkowicz, Jay Gopalakrishnan, Ignacio Muga, Jeffrey Zitelli Jan 2012

Wavenumber Explicit Analysis Of A Dpg Method For The Multidimensional Helmholtz Equation, Leszek Demkowicz, Jay Gopalakrishnan, Ignacio Muga, Jeffrey Zitelli

Mathematics and Statistics Faculty Publications and Presentations

We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order case. Theoretically, we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber ω, the mesh size h, and the polynomial degree p. But the current state of the theory does not fully explain the remarkably good numerical phase errors. Theoretically, comparisons are made with …

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The Evil Technology Hypothesis: A Deep Ecological Reading Of International Law, Ugo Mattei, Luigi Russi Jan 2012

The Evil Technology Hypothesis: A Deep Ecological Reading Of International Law, Ugo Mattei, Luigi Russi

Faculty Scholarship

No abstract provided.

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A Class Of Discontinuous Petrov–Galerkin Methods. Part Iii: Adaptivity, Leszek Demkowicz, Jay Gopalakrishnan, Antti H. Niemi Jan 2012

A Class Of Discontinuous Petrov–Galerkin Methods. Part Iii: Adaptivity, Leszek Demkowicz, Jay Gopalakrishnan, Antti H. Niemi

Mathematics and Statistics Faculty Publications and Presentations

We continue our theoretical and numerical study on the Discontinuous Petrov–Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: for 1D and for 2D problems. The adaptive process is fully automatic and starts …

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Recollecting Turbulence: Catastrophe And Sacrifice In The "History Of My Life" By Henry Darger, Carl Watson Jan 2012

Recollecting Turbulence: Catastrophe And Sacrifice In The "History Of My Life" By Henry Darger, Carl Watson

Dissertations, Theses, and Capstone Projects

This study of "The History of My Life" the 5,086 page autobiographical text by the outsider artist/author Henry Darger, uses non-linear modes of analysis, such as chaos and complexity theory, to explore the meaning of Darger's epic narrative. Beginning with the idea that turbulence, seemingly chaotic, actually comes about as a compensatory restructuring of inadequate or unstable system dynamics, this study goes on to show that, as both influence and effect, turbulence is found at every level of Darger's life and art, both in theme and structure. "My Life" is a prime example: an extended narrative describing a cataclysmic tornado, …

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Civilian : Civil Engineering Newsletter : Fall 2012 Jan 2012

Civilian : Civil Engineering Newsletter : Fall 2012

Civilian Newsletter

No abstract provided.

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Optomechanics Of Cavity Driven Nanoparticles, Joel T. Rubin Jan 2012

Optomechanics Of Cavity Driven Nanoparticles, Joel T. Rubin

Dissertations, Theses, and Capstone Projects

The subject of this thesis is the opto-mechanical interaction of a spherical high-Q microresonator and a subwavelength particle, which, at optical wavelengths, corresponds to a size on the order of nanometers. After a review of the basic theory of spherical resonators and multi-sphere scattering, the full self-consistent electromagnetic field of the coupled resonator-particle system is derived. The particle-induced frequency shift and broadening is calculated by examining the poles of the scattering coefficients of the resonator. The force exerted on the particle by the field is determined via the Maxwell stress tensor, and is found to be in general non-conservative. From …

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