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Articles 1 - 17 of 17
Full-Text Articles in Physical Sciences and Mathematics
One-Sided Derivative Of Distance To A Compact Set, Logan S. Fox, Peter Oberly, J. J. P. Veerman
One-Sided Derivative Of Distance To A Compact Set, Logan S. Fox, Peter Oberly, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
We give a complete and self-contained proof of a folklore theorem which says that in an Alexandrov space the distance between a point γ(t) on a geodesic γ and a compact set K is a right-differentiable function of t. Moreover, the value of this right-derivative is given by the negative cosine of the minimal angle between the geodesic and any shortest path to the compact set (Theorem 4.3). Our treatment serves as a general introduction to metric geometry and relies only on the basic elements, such as comparison triangles and upper angles.
Spacetime Numerical Techniques For The Wave And Schrödinger Equations, Paulina Ester Sepùlveda Salas
Spacetime Numerical Techniques For The Wave And Schrödinger Equations, Paulina Ester Sepùlveda Salas
Dissertations and Theses
The most common tool for solving spacetime problems using finite elements is based on semidiscretization: discretizing in space by a finite element method and then advancing in time by a numerical scheme. Contrary to this standard procedure, in this dissertation we consider formulations where time is another coordinate of the domain. Therefore, spacetime problems can be studied as boundary value problems, where initial conditions are considered as part of the spacetime boundary conditions.
When seeking solutions to these problems, it is natural to ask what are the correct spaces of functions to choose, to obtain wellposedness. This motivates the study …
A Parallel Mesh Generator In 3d/4d, Kirill Voronin
A Parallel Mesh Generator In 3d/4d, Kirill Voronin
Portland Institute for Computational Science Publications
In the report a parallel mesh generator in 3d/4d is presented. The mesh generator was developed as a part of the research project on space-time discretizations for partial differential equations in the least-squares setting. The generator is capable of constructing meshes for space-time cylinders built on an arbitrary 3d space mesh in parallel. The parallel implementation was created in the form of an extension of the finite element software MFEM. The code is publicly available in the Github repository
High-Order Method For Evaluating Derivatives Of Harmonic Functions In Planar Domains, Jeffrey S. Ovall, Samuel E. Reynolds
High-Order Method For Evaluating Derivatives Of Harmonic Functions In Planar Domains, Jeffrey S. Ovall, Samuel E. Reynolds
Mathematics and Statistics Faculty Publications and Presentations
We propose a high-order integral equation based method for evaluating interior and boundary derivatives of harmonic functions in planar domains that are specified by their Dirichlet data.
Equators Have At Most Countable Many Singularities With Bounded Total Angle, Pilar Herreros, Mario Ponce, J.J.P. Veerman
Equators Have At Most Countable Many Singularities With Bounded Total Angle, Pilar Herreros, Mario Ponce, J.J.P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix Lpq as the set of equidistant points to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. In the case of a topological sphere, mediatrices are called equators and it can benoticed that there are no branching points, thus an equator is a topological circle with possibly many Lipschitz singularities. This paper establishes that mediatrices have the radial …
Transients In The Synchronization Of Asymmetrically Coupled Oscillator Arrays, Carlos E. Cantos, David K. Hammond, J.J.P. Veerman
Transients In The Synchronization Of Asymmetrically Coupled Oscillator Arrays, Carlos E. Cantos, David K. Hammond, J.J.P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
We consider the transient behavior of a large linear array of coupled linear damped harmonic oscillators following perturbation of a single element. Our work is motivated by modeling the behavior of flocks of autonomous vehicles. We first state a number of conjectures that allow us to derive an explicit characterization of the transients, within a certain parameter regime Ω. As corollaries we show that minimizing the transients requires considering non-symmetric coupling, and that within Ω the computed linear growth in N of the transients is independent of (reasonable) boundary conditions.
On The Coupling Of Dpg And Bem, Thomas Führer, Norbert Heuer, Michael Karkulik
On The Coupling Of Dpg And Bem, Thomas Führer, Norbert Heuer, Michael Karkulik
Mathematics and Statistics Faculty Publications and Presentations
We develop and analyze strategies to couple the discontinuous Petrov-Galerkin method with optimal test functions to (i) least-squares boundary elements and (ii) various variants of standard Galerkin boundary elements. An essential feature of our method is that, despite the use of boundary integral equations, optimal test functions have to be computed only locally. We apply our findings to a standard transmission problem in full space and present numerical experiments to validate our theory.
Regularity Of Mediatrices In Surfaces, Pilar Herreros, Mario Ponce, J. J. P. Veerman
Regularity Of Mediatrices In Surfaces, Pilar Herreros, Mario Ponce, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix Lpq as the set of equidistant points to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes additional geometric regularity properties of mediatrices. We show that mediatrices have the radial linearizability property, which implies that at each point they have a geometrically defined derivative in the branching directions. Also, we study the particular case of mediatrices on spheres, by showing that they are Lipschitz simple closed curves …
Well Conditioned Boundary Integral Equations For Two-Dimensional Sound-Hard Scattering Problems In Domains With Corners, Akash Anand, Jeffrey S. Ovall, Catalin Turc
Well Conditioned Boundary Integral Equations For Two-Dimensional Sound-Hard Scattering Problems In Domains With Corners, Akash Anand, Jeffrey S. Ovall, Catalin Turc
Mathematics and Statistics Faculty Publications and Presentations
We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in domains with corners. We focus mainly on Direct Regularized Combined Field Integral Equation (DCFIE-R) formulations whose name reflects that (1) they consist of combinations of direct boundary integral equations of the second-kind and first-kind integral equations which are preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are physical quantities, i.e., the total field on the boundary of the scatterer. The DCFIE-R equations are shown to be uniquely solvable in appropriate function …
Partial Expansion Of A Lipschitz Domain And Some Applications, Weifeng Qiu, Jay Gopalakrishnan
Partial Expansion Of A Lipschitz Domain And Some Applications, Weifeng Qiu, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated
Reliable A-Posteriori Error Estimators For Hp-Adaptive Finite Element Approximations Of Eigenvalue/Leigenvector Problems, Stefano Giani, Luka Grubisic, Jeffrey S. Ovall
Reliable A-Posteriori Error Estimators For Hp-Adaptive Finite Element Approximations Of Eigenvalue/Leigenvector Problems, Stefano Giani, Luka Grubisic, Jeffrey S. Ovall
Mathematics and Statistics Faculty Publications and Presentations
We present reliable a-posteriori error estimates for hp-adaptive finite element approxima- tions of eigenvalue/eigenvector problems. Starting from our earlier work on h adaptive finite element approximations we show a way to obtain reliable and efficient a-posteriori estimates in the hp-setting. At the core of our analysis is the reduction of the problem on the analysis of the associated boundary value problem. We start from the analysis of Wohlmuth and Melenk and combine this with our a-posteriori estimation framework to obtain eigenvalue/eigenvector approximation bounds.
A Second Elasticity Element Using The Matrix Bubble, Jay Gopalakrishnan, Johnny Guzmán
A Second Elasticity Element Using The Matrix Bubble, Jay Gopalakrishnan, Johnny Guzmán
Mathematics and Statistics Faculty Publications and Presentations
We presented a family of finite elements that use a polynomial space augmented by certain matrix bubbles in Cockburn et al. (2010) A new elasticity element made for enforcing weak stress symmetry. Math. Comput., 79, 1331–1349 . In this sequel we exhibit a second family of elements that use the same matrix bubble. This second element uses a stress space smaller than the first while maintaining the same space for rotations (which are the Lagrange multipliers corresponding to a weak symmetry constraint). The space of displacements is of one degree less than the first method. The analysis, while similar to …
Green Dyadic For The Proca Fields, P.T. Leung, Paul Dragulin
Green Dyadic For The Proca Fields, P.T. Leung, Paul Dragulin
Physics Faculty Publications and Presentations
The dyadic Green functions for the Proca fields in free space are derived to include singular terms. Both the electric and magnetic types will be obtained with the results reduced back to those for the Maxwell fields in the limit of zero photon mass. Moreover, the singular terms are identical in both massless and massive electrodynamics. As an illustration, the results are applied to obtain the exact dynamical fields for an oscillating dipole which reduce back to the well-known expressions for static fields derived previously in the literature for massive electrodynamics.
On The Spectra Of Certain Directed Paths, Carlos Martins Da Fonseca, J. J. P. Veerman
On The Spectra Of Certain Directed Paths, Carlos Martins Da Fonseca, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
We describe the eigenpairs of special kinds of tridiagonal matrices related to problems on traffic on a one-lane road. Some numerical examples are provided.
Incompressible Finite Elements Via Hybridization. Part I: The Stokes System In Two Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan
Incompressible Finite Elements Via Hybridization. Part I: The Stokes System In Two Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
In this paper, we introduce a new and efficient way to compute exactly divergence-free velocity approximations for the Stokes equations in two space dimensions. We begin by considering a mixed method that provides an exactly divergence-free approximation of the velocity and a continuous approximation of the vorticity. We then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique. This novel formulation bypasses the difficult task of constructing an exactly divergence-free basis for velocity approximations. Moreover, the discrete system resulting from our method has fewer degrees …
Incompressible Finite Elements Via Hybridization. Part Ii: The Stokes System In Three Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan
Incompressible Finite Elements Via Hybridization. Part Ii: The Stokes System In Three Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
We introduce a method that gives exactly incompressible velocity approximations to Stokes ow in three space dimensions. The method is designed by extending the ideas in Part I (http://archives.pdx.edu/ds/psu/10914) of this series, where the Stokes system in two space dimensions was considered. Thus we hybridize a vorticity-velocity formulation to obtain a new mixed method coupling approximations of tangential velocity and pressure on mesh faces. Once this relatively small tangential velocity-pressure system is solved, it is possible to recover a globally divergence-free numerical approximation of the fluid velocity, an approximation of the vorticity whose tangential component is continuous across …
Molecular Lifetimes In The Presence Of Periodically Roughened Metallic Surfaces, P.T. Leung, Z. C. Wu, Daniel A. Jelski, Thomas F. George
Molecular Lifetimes In The Presence Of Periodically Roughened Metallic Surfaces, P.T. Leung, Z. C. Wu, Daniel A. Jelski, Thomas F. George
Physics Faculty Publications and Presentations
The lifetimes of molecules located close to a sinusoidal grating surface are studied within a classical phenomenological model. The contribution of surface roughness to the molecular decay rate is attributed to the discrepancy between the experiments of Rossetti and Brus and the theory of Chance, Prock, and Silbey. It is found that surface roughness can either enhance or diminish the flat-surface value for the decay rate depending on the emitting frequency, molecule-surface distance, and the molecular orientation.