Dynamics Of Locally Coupled Oscillators With Next-Nearest-Neighbor Interaction, 2016 University of Crete

#### Dynamics Of Locally Coupled Oscillators With Next-Nearest-Neighbor Interaction, J. Herbrych, A. G. Chazirakis, N. Christakis, J.J. P. Veerman

*J.J.P. Veerman*

A theoretical description of decentralized dynamics within linearly coupled, one-dimensional oscillators (agents) with up to next-nearest-neighbor interaction is given. Conditions for stability of such system are presented. Our results indicate that the stable systems have response that grow at least linearly in the system size. We give criteria when this is the case. The dynamics of these systems can be described with traveling waves with strong damping in the high frequencies. Depending on the system parameters, two types of solutions have been found: damped oscillations and reflectionless waves. The latter is a novel result and a feature of systems with ...

Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, 2016 University of Minnesota - Twin Cities

#### Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan, Raytcho Lazarov

*Jay Gopalakrishnan*

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 ...

The Derivation Of Hybridizable Discontinuous Galerkin Methods For Stokes Flow, 2016 University of Minnesota - Twin Cities

#### The Derivation Of Hybridizable Discontinuous Galerkin Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan

*Jay Gopalakrishnan*

In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.

The Convergence Of V-Cycle Multigrid Algorithms For Axisymmetric Laplace And Maxwell Equations, 2016 Portland State University

#### The Convergence Of V-Cycle Multigrid Algorithms For Axisymmetric Laplace And Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity ...

Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, 2016 Portland State University

#### Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

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 ...

Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics, 2016 Portland State University

#### Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics, Dylan M. Copeland, Jay Gopalakrishnan, Minah Oh

*Jay Gopalakrishnan*

Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex then the multigrid Vcycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for ...

Multigrid For The Mortar Finite Element Method, 2016 Portland State University

#### Multigrid For The Mortar Finite Element Method, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

A multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed. These problems are posed on domains partitioned into subdomains, each of which is independently triangulated in a multilevel fashion. The multilevel mortar finite element spaces based on such triangulations (which need not align across subdomain interfaces) are in general not nested. Suitable grid transfer operators and smoothers are developed which lead to a variable Vcycle preconditioner resulting in a uniformly preconditioned algebraic system. Computational results illustrating the theory are also presented.

Nédélec Spaces In Affine Coordinates, 2016 Portland State University

#### Nédélec Spaces In Affine Coordinates, Jay Gopalakrishnan, Luis E. García-Castillo, Leszek Demkowicz

*Jay Gopalakrishnan*

In this note, we provide a conveniently implementable basis for simplicial Nédélec spaces of any order in any space dimension. The main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex.

Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, 2016 Portland State University

#### Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, Jay Gopalakrishnan, Johnny Guzmán

*Jay Gopalakrishnan*

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 ...

Polynomial Extension Operators. Part I, 2016 University of Texas at Austin

#### Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

*Jay Gopalakrishnan*

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 H^{½}(∂K) into H¹(K), for any tetrahedron K.

Quasioptimality Of Some Spectral Mixed Methods, 2016 Portland State University

#### Quasioptimality Of Some Spectral Mixed Methods, Jay Gopalakrishnan, Leszek Demkowicz

*Jay Gopalakrishnan*

In this paper, we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree. Using the projectors, we obtain quasioptimality of some spectralmixed methods, including the Raviart–Thomas method and mixed formulations of Maxwell equations. We also prove some discrete Friedrichs type inequalities involving curl.

Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, 2016 University of Minnesota - Twin Cities

#### Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, Bernardo Cockburn, Jay Gopalakrishnan

*Jay Gopalakrishnan*

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 ...

Hybridization And Postprocessing Techniques For Mixed Eigenfunctions, 2016 University of Minnesota - Twin Cities

#### Hybridization And Postprocessing Techniques For Mixed Eigenfunctions, Bernardo Cockburn, Jay Gopalakrishnan, F. Li, Ngoc Cuong Nguyen, Jaume Peraire

*Jay Gopalakrishnan*

We introduce hybridization and postprocessing techniques for the Raviart–Thomas approximation of second-order elliptic eigenvalue problems. Hybridization reduces the Raviart–Thomas approximation to a condensed eigenproblem. The condensed eigenproblem is nonlinear, but smaller than the original mixed approximation. We derive multiple iterative algorithms for solving the condensed eigenproblem and examine their interrelationships and convergence rates. An element-by-element postprocessing technique to improve accuracy of computed eigenfunctions is also presented. We prove that a projection of the error in the eigenspace approximation by the mixed method (of any order) superconverges and that the postprocessed eigenfunction approximations converge faster for smooth eigenfunctions. Numerical ...

Mortar Estimates Independent Of Number Of Subdomains, 2016 Portland State University

#### Mortar Estimates Independent Of Number Of Subdomains, Jay Gopalakrishnan

*Jay Gopalakrishnan*

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.

Incompressible Finite Elements Via Hybridization. Part Ii: The Stokes System In Three Space Dimensions, 2016 University of Minnesota - Twin Cities

#### Incompressible Finite Elements Via Hybridization. Part Ii: The Stokes System In Three Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan

*Jay Gopalakrishnan*

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 ...

A Second Elasticity Element Using The Matrix Bubble, 2016 Portland State University

#### A Second Elasticity Element Using The Matrix Bubble, Jay Gopalakrishnan, Johnny Guzmán

*Jay Gopalakrishnan*

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 ...

An Efficient Method For Band Structure Calculations In 3d Photonic Crystals, 2016 Texas A & M University - College Station

#### An Efficient Method For Band Structure Calculations In 3d Photonic Crystals, David C. Dobson, Jay Gopalakrishnan, Joseph E. Pasciak

*Jay Gopalakrishnan*

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.

A Tent Pitching Scheme Motivated By Friedrichs Theory, 2016 Portland State University

#### A Tent Pitching Scheme Motivated By Friedrichs Theory, Jay Gopalakrishnan, Peter Monk, Paulina Sepúlveda

*Jay Gopalakrishnan*

Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet. Motivated by this continuity property, an explicit space-time finite element scheme of the tent pitching type, with spaces that conform to the continuity property, is designed. Numerical results for a model one-dimensional wave propagation problem are presented.

Analysis Of The Dpg Method For The Poisson Equation, 2016 University of Texas at Austin

#### Analysis Of The Dpg Method For The Poisson Equation, Leszek Demkowicz, Jay Gopalakrishnan

*Jay Gopalakrishnan*

We give an error analysis of the recently developed DPG method applied to solve the Poisson equation and a convection-dffusion problem. We prove that the method is quasioptimal. Error estimates in terms of both the mesh size h and the polynomial degree p (for various element shapes) can be derived from our results. Results of extensive numerical experiments are also presented.

Asymptotic And Numerical Techniques For Resonances Of Thin Photonic Structures, 2016 Portland State University

#### Asymptotic And Numerical Techniques For Resonances Of Thin Photonic Structures, Jay Gopalakrishnan, Shari Moskow, Fadil Santosa

*Jay Gopalakrishnan*

We consider the problem of calculating resonance frequencies and radiative losses of an optical resonator. The optical resonator is in the form of a thin membrane with variable dielectric properties. This work provides two very different approaches for doing such calculations. The first is an asymptotic method which exploits the small thickness and high index of the membrane. We derive a limiting resonance problem as the thickness goes to zero, and for the case of a simple resonance, find a first order correction. The limiting problem and the correction are in one less space dimension, which can make the approach ...