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- Mathematics and Statistics Faculty Publications and Presentations (23)
- Mathematics Research Reports (7)
- UNLV Theses, Dissertations, Professional Papers, and Capstones (5)
- Mathematics and Statistics Faculty Research & Creative Works (3)
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Articles 31 - 56 of 56
Full-Text Articles in Physical Sciences and Mathematics
Leonhard Euler's Contribution To Infinite Polynomials, Jack Dean Meekins
Leonhard Euler's Contribution To Infinite Polynomials, Jack Dean Meekins
Theses Digitization Project
This thesis will focus on Euler's famous method for solving the infinite polynomial. It will show how he manipulated the sine function to find all possible points along the sine function such that the sine A would equal to y; these would be roots of the polynomial. It also shows how Euler set the infinite polynomial equal to the infinite product allowing him to determine which coefficients were equal to which reciprocals of the roots, roots squared, roots cubed, etc.
Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, Jay Gopalakrishnan, Johnny Guzmán
Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, Jay Gopalakrishnan, Johnny Guzmán
Mathematics and Statistics Faculty Publications and Presentations
We present a family of mixed methods for linear elasticity that yield exactly symmetric, but only weakly conforming, stress approximations. The method is presented in both two and three dimensions (on triangular and tetrahedral meshes). The method is efficiently implementable by hybridization. The degrees of freedom of the Lagrange multipliers, which approximate the displacements at the faces, solve a symmetric positive-definite system. The design and analysis of this method is motivated by a new set of unisolvent degrees of freedom for symmetric polynomial matrices. These new degrees of freedom are also used to give a new simple calculation of the …
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 …
Commuting Smoothed Projectors In Weighted Norms With An Application To Axisymmetric Maxwell Equations, Jay Gopalakrishnan, Minah Oh
Commuting Smoothed Projectors In Weighted Norms With An Application To Axisymmetric Maxwell Equations, Jay Gopalakrishnan, Minah Oh
Mathematics and Statistics Faculty Publications and Presentations
We construct finite element projectors that can be applied to functions with low regularity. These projectors are continuous in a weighted norm arising naturally when modeling devices with axial symmetry. They have important commuting diagram properties needed for finite element analysis. As an application, we use the projectors to prove quasioptimal convergence for the edge finite element approximation of the axisymmetric time-harmonic Maxwell equations on nonsmooth domains. Supplementary numerical investigations on convergence deterioration at high wavenumbers and near Maxwell eigenvalues and are also reported.
A Class Of Discontinuous Petrov–Galerkin Methods. Part Iv: The Optimal Test Norm And Time-Harmonic Wave Propagation In 1d., Jeffrey Zitelli, Leszek Demkowicz, Jay Gopalakrishnan, D. Pardo, V. M. Calo
A Class Of Discontinuous Petrov–Galerkin Methods. Part Iv: The Optimal Test Norm And Time-Harmonic Wave Propagation In 1d., Jeffrey Zitelli, Leszek Demkowicz, Jay Gopalakrishnan, D. Pardo, V. M. Calo
Mathematics and Statistics Faculty Publications and Presentations
The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test …
Numerical Computations For Pde Models Of Rocket Exhaust Flow In Soil, Brian Brennan
Numerical Computations For Pde Models Of Rocket Exhaust Flow In Soil, Brian Brennan
Electronic Theses and Dissertations
We study numerical methods for solving the nonlinear porous medium and Navier-Lame problems. When coupled together, these equations model the flow of exhaust through a porous medium, soil, and the effects that the pressure has on the soil in terms of spatial displacement. For the porous medium equation we use the Crank-Nicolson time stepping method with a spectral discretization in space. Since the Navier-Lame equation is a boundary value problem, it is solved using a finite element method where the spatial domain is represented by a triangulation of discrete points. The two problems are coupled by using approximations of solutions …
A Class Of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation, Leszek Demkowicz, Jay Gopalakrishnan
A Class Of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation, Leszek Demkowicz, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size h and polynomial degree p, and outperforms the standard upwind DG method.
Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan, Raytcho Lazarov
Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan, Raytcho Lazarov
Mathematics and Statistics Faculty Publications and Presentations
We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continu- ous Galerkin, non-conforming and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric and positive definite, these methods can be efficiently implemented. Moreover, the …
Polynomial Extension Operators. Part Ii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl
Polynomial Extension Operators. Part Ii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl
Mathematics and Statistics Faculty Publications and Presentations
Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such extensions are a fundamental ingredient of high order finite element analysis.
Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak
Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak
Mathematics and Statistics Faculty Publications and Presentations
The finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither Hermitian nor definite when the coefficient function is complex valued. For such problems, under a uniqueness assumption, we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero. Then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number, providing an efficient solver for computing discrete approximations. Numerical experiments, while confirming the theory, …
Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl
Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl
Mathematics and Statistics Faculty Publications and Presentations
In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H½(∂K) into H¹(K), for any tetrahedron K.
Locally Conservative Fluxes For The Continuous Galerkin Method, Bernardo Cockburn, Jay Gopalakrishnan, Haiying Wang
Locally Conservative Fluxes For The Continuous Galerkin Method, Bernardo Cockburn, Jay Gopalakrishnan, Haiying Wang
Mathematics and Statistics Faculty Publications and Presentations
The standard continuous Galerkin (CG) finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations, a much needed quantity in many applications. We show how to overcome this shortcoming by using a two step postprocessing. The first step is the computation of a numerical flux trace defined on element inter- faces and is motivated by the structure of the numerical traces of discontinuous Galerkin methods. This computation is non-local in that it requires the solution of a symmetric positive definite system, but the system is well conditioned independently of mesh size, so it …
Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang
Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang
Mathematics Research Reports
We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two dimensional Poisson equation. Results of this paper are two folds. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered.gradient by the Polynomial Preserving Recovery (PPR) is superconvergent. Secondly, we demonstrate by numerical examples that an adaptive procedure with a posteriori error estimator based on PPR does produce adaptive meshes satisfy our mesh density assumption, and the recovered gradient by PPR is indeed supercoveregent in the adaptive process.
Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, Bernardo Cockburn, Jay Gopalakrishnan
Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, Bernardo Cockburn, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new …
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 …
Natural Superconvergent Points Of Triangular Finite Elements, Zhimin Zhang, Runchang Lin
Natural Superconvergent Points Of Triangular Finite Elements, Zhimin Zhang, Runchang Lin
Mathematics Research Reports
In this work, we analytically identify natural superconvergent points of function values and gradients for triangular elements. Both the Poisson equation and the Laplace equation are discussed for polynomial finite element spaces (with degrees up to 8) under four different mesh patterns. Our results verify computer findings of [2], especially, we confirm that the computed data have 9 digits of accuracy with an exception of one pair (which has 8-7 digits of accuracy). In addition, we demonstrate that the function value superconvergent points predicted by the symmetry theory [14] are the only superconvergent points for the Poisson equation. Finally, we …
Validation Of The A Posteriori Error Estimator Based On Polynomial Preserving Recovery For Linear Elements, Zhimin Zhang, Ahmed Naga
Validation Of The A Posteriori Error Estimator Based On Polynomial Preserving Recovery For Linear Elements, Zhimin Zhang, Ahmed Naga
Mathematics Research Reports
In this paper the quality of the error estimator based on the Polynomial Preserving Recovery (PPR) is investigated using the computer-based approach proposed by Babiiska et al. A comparison is made between the error estimator based on the PPR and the one based on the Superconvergence Patch Recovery (SPR). It was found that the PPR is at least as good as the SPR.
A Schwarz Preconditioner For A Hybridized Mixed Method, Jay Gopalakrishnan
A Schwarz Preconditioner For A Hybridized Mixed Method, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
In this paper, we provide a Schwarz preconditioner for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods. The preconditioner is for the linear equation for Lagrange multipliers arrived at by eliminating the ux as well as the primal variable. We also prove a condition number estimate for this equation when no preconditioner is used. Although preconditioners for the lowest order case of the Raviart-Thomas method have been constructed previously by exploiting its connection with a nonconforming method, our approach is different, in that we use a new variational characterization of the Lagrange multiplier equation. This allows us to …
A Multilevel Discontinuous Galerkin Method, Jay Gopalakrishnan, Guido Kanschat
A Multilevel Discontinuous Galerkin Method, Jay Gopalakrishnan, Guido Kanschat
Mathematics and Statistics Faculty Publications and Presentations
A variable V-cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented. An analysis under a mild regularity assumption shows that the preconditioner is uniform. The interior penalty method is then combined with a discontinuous Galerkin scheme to arrive at a discretization scheme for an advection-diffusion problem, for which an error estimate is proved. A multigrid algorithm for this method is presented, and numerical experiments indicating its robustness with respect to diffusion coefficient are reported.
A Posteriori Error Estimates Based On Polynomial Preserving Recovery, Zhimin Zhang, Ahmed Naga
A Posteriori Error Estimates Based On Polynomial Preserving Recovery, Zhimin Zhang, Ahmed Naga
Mathematics Research Reports
Superconvergence of order O(h1+rho), for some rho is greater than 0, is established for gradients recovered using Polynomial Preserving Recovery technique when the mesh is mildly structured. Consequently this technique can be used in building a posteriori error estimator that is asymptotically exact.
Gradient Recovery And A Posteriori Estimate For Bilinear Element On Irregular Quadrilateral Meshes, Zhimin Zhang
Gradient Recovery And A Posteriori Estimate For Bilinear Element On Irregular Quadrilateral Meshes, Zhimin Zhang
Mathematics Research Reports
A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under general quadrilateral meshes. It has been proven that the recovered gradient converges at a rate O(h1+rho) for rho = min(alpha, 1) when the mesh is distorted O(h1+alpha) (alpha > 0) from a regular one. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact.
A Meshless Gradient Recovery Method Part I: Superconvergence Property, Zhiming Zhang, Ahmed Naga
A Meshless Gradient Recovery Method Part I: Superconvergence Property, Zhiming Zhang, Ahmed Naga
Mathematics Research Reports
A new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz-Zhu patch recovery method. In addition, under uniform triangular meshes, the method is superconvergent for the Chevron pattern, and ultraconvergence at element edge centers for the regular pattern.
Ultraconvergence Of Zz Patch Recovery At Mesh Symmetry Points, Zhimin Zhang, Runchang Lin
Ultraconvergence Of Zz Patch Recovery At Mesh Symmetry Points, Zhimin Zhang, Runchang Lin
Mathematics Research Reports
Ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed. Superconvergence recovery for the Q8 element is proved and ultraconvergence numerical examples are demonstrated.
An Efficient Method For Band Structure Calculations In 3d Photonic Crystals, David C. Dobson, Jay Gopalakrishnan, Joseph E. Pasciak
An Efficient Method For Band Structure Calculations In 3d Photonic Crystals, David C. Dobson, Jay Gopalakrishnan, Joseph E. Pasciak
Mathematics and Statistics Faculty Publications and Presentations
A method for computing band structures for three-dimensional photonic crystals is described. The method combines a mixed finite element discretization on a uniform grid with a fast Fourier transform preconditioner and a preconditioned subspace iteration algorithm. Numerical examples illustrating the behavior of the method are presented.
Mortar Estimates Independent Of Number Of Subdomains, Jay Gopalakrishnan
Mortar Estimates Independent Of Number Of Subdomains, Jay Gopalakrishnan
Mathematics and Statistics Faculty Publications and Presentations
The stability and error estimates for the mortar finite element method are well established. This work examines the dependence of constants in these estimates on shape and number of subdomains. By means of a Poincar´e inequality and some scaling arguments, these estimates are found not to deteriorate with increase in number of subdomains.