arXiv subjects

Jay Gopalakrishnan

Publications and source records attributed to Jay Gopalakrishnan.

At least 37 records · Page 2Linked to original sources

arXiv2021

Computing leaky modes of optical fibers using a FEAST algorithm for polynomial eigenproblems

An efficient contour integral technique to approximate a cluster of nonlinear eigenvalues of a polynomial eigenproblem, circumventing certain large inversions from a linearization, is presented. It is applied to the nonlinear eigenproblem that arises from a frequency-dependent perfectly matched layer. This approach is shown to result in an accurate method for computing leaky modes of optical fibers. Extensive computations on an antiresonant fiber with a complex transverse microstructure are reported. This structure is found to present substantial computational difficulties: Even when employing over one million degrees of freedom, the fiber model appears to remain in a preasymptotic regime where computed confinement loss values are likely to be off by orders of magnitude. Other difficulties in computing mode losses, together with practical techniques to overcome them, are detailed.

physics.optics↗

arXiv2021

Convergence analysis of some tent-based schemes for linear hyperbolic systems

Finite element methods for symmetric linear hyperbolic systems using unstructured advancing fronts (satisfying a causality condition) are considered in this work. Convergence results and error bounds are obtained for mapped tent pitching schemes made with standard discontinuous Galerkin discretizations for spatial approximation on mapped tents. Techniques to study semidiscretization on mapped tents, design fully discrete schemes, prove local error bounds, prove stability on spacetime fronts, and bound error propagated through unstructured layers are developed.

math.NA↗

arXiv2020

A discrete elasticity complex on three-dimensional Alfeld splits

We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de~Rham complexes, and smoother finite element differential forms.

math.NA↗

arXiv2020

Structure aware Runge-Kutta time stepping for spacetime tents

We introduce a new class of Runge-Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge-Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.

math.NA↗

arXiv2019

An Explicit Mapped Tent Pitching Scheme for Maxwell Equations

We present a new numerical method for solving time dependent Maxwell equations, which is also suitable for general linear hyperbolic equations. It is based on an unstructured partitioning of the spacetime domain into tent-shaped regions that respect causality. Provided that an approximate solution is available at the tent bottom, the equation can be locally evolved up to the top of the tent. By mapping tents to a domain which is a tensor product of a spatial domain with a time interval, it is possible to construct a fully explicit scheme that advances the solution through unstructured meshes. This work highlights a difficulty that arises when standard explicit Runge Kutta schemes are used in this context and proposes an alternative structure-aware Taylor time-stepping technique. Thus explicit methods are constructed that allow variable time steps and local refinements without compromising high order accuracy in space and time. These Mapped Tent Pitching (MTP) schemes lead to highly parallel algorithms, which utilize modern computer architectures extremely well.

math.NA↗

arXiv2019

Simulation of Optical Fiber Amplifier Gain Using Equivalent Short Fibers

Electromagnetic wave propagation in optical fiber amplifiers obeys Maxwell equations. Using coupled mode theory, the full Maxwell system within an optical fiber amplifier is reduced to a simpler model. The simpler model is made more efficient through a new scale model, referred to as an equivalent short fiber, which captures some of the essential characteristics of a longer fiber. The equivalent short fiber can be viewed as a fiber made using artificial (unphysical) material properties that in some sense compensates for its reduced length. The computations can be accelerated by a factor approximately equal to the ratio of the original length to the reduced length of the equivalent fiber. Computations using models of two commercially available fibers -- one doped with ytterbium, and the other with thulium -- show the practical utility of the concept. Extensive numerical studies are conducted to assess when the equivalent short fiber model is useful and when it is not.

math.NA↗

arXiv2019

A mass conserving mixed stress formulation for Stokes flow with weakly imposed stress symmetry

We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses. The method is built upon a mass conserving mixed formulation that we recently studied. The improvement in this work is a new method that directly approximates the viscous fluid stress $σ$, enforcing its symmetry weakly. The finite element space in which the stress is approximated consists of matrix-valued functions having continuous "normal-tangential" components across element interfaces. Stability is achieved by adding certain matrix bubbles that were introduced earlier in the literature on finite elements for linear elasticity. Like the earlier work, the new method here approximates the fluid velocity $u$ using $H(\operatorname{div})$-conforming finite elements, thus providing exact mass conservation. Our error analysis shows optimal convergence rates for the pressure and the stress variables. An additional post processing yields an optimally convergent velocity satisfying exact mass conservation. The method is also pressure robust.

math.NA↗

arXiv2019

Analysis of FEAST spectral approximations using the DPG discretization

A filtered subspace iteration for computing a cluster of eigenvalues and its accompanying eigenspace, known as "FEAST", has gained considerable attention in recent years. This work studies issues that arise when FEAST is applied to compute part of the spectrum of an unbounded partial differential operator. Specifically, when the resolvent of the partial differential operator is approximated by the discontinuous Petrov Galerkin (DPG) method, it is shown that there is no spectral pollution. The theory also provides bounds on the discretization errors in the spectral approximations. Numerical experiments for simple operators illustrate the theory and also indicate the value of the algorithm beyond the confines of the theoretical assumptions. The utility of the algorithm is illustrated by applying it to compute guided transverse core modes of a realistic optical fiber.

math.NA↗

arXiv2019

Spectral discretization errors in filtered subspace iteration

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates.

math.NA↗

arXiv2018

A spacetime DPG method for the wave equation in multiple dimensions

A spacetime discontinuous Petrov-Galerkin (DPG) method for the linear wave equation is presented. This method is based on a weak formulation that uses a broken graph space. The wellposedness of this formulation is established using a previously presented abstract framework. One of the main tasks in the verification of the conditions of this framework is proving a density result. This is done in detail for a simple domain in arbitrary dimensions. The DPG method based on the weak formulation is then studied theoretically and numerically. Error estimates and numerical results are presented for triangular, rectangular, tetrahedral, and hexahedral meshes of the spacetime domain. The potential for using the built-in error estimator of the DPG method for an adaptivity mesh refinement strategy in two and three dimensions is also presented.

math.NA↗

arXiv2018

The DPG-star method

This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG methods can be seen as generalizations of $\mathcal{L}\mathcal{L}^\ast$ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG$^*$ method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG$^*$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.

math.NA↗

arXiv2018

A mass conserving mixed stress formulation for the Stokes equations

We propose a new discretization of a mixed stress formulation of the Stokes equations. The velocity $u$ is approximated with $H(\operatorname{div})$-conforming finite elements providing exact mass conservation. While many standard methods use $H^1$-conforming spaces for the discrete velocity, $H(\operatorname{div})$-conformity fits the considered variational formulation in this work. A new stress-like variable $σ$ equalling the gradient of the velocity is set within a new function space $H(\operatorname{curl} \operatorname{div})$. New matrix-valued finite elements having continuous "normal-tangential" components are constructed to approximate functions in $H(\operatorname{curl} \operatorname{div})$. An error analysis concludes with optimal rates of convergence for errors in $u$ (measured in a discrete $H^1$-norm), errors in $σ$ (measured in $L^2$) and the pressure $p$ (also measured in $L^2$). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.

math.NA↗

arXiv2018

The auxiliary space preconditioner for the de Rham complex

We generalize the construction and analysis of auxiliary space preconditioners to the n-dimensional finite element subcomplex of the de Rham complex. These preconditioners are based on a generalization of a decomposition of Sobolev space functions into a regular part and a potential. A discrete version is easily established using the tools of finite element exterior calculus. We then discuss the four-dimensional de Rham complex in detail. By identifying forms in four dimensions (4D) with simple proxies, form operations are written out in terms of familiar algebraic operations on matrices, vectors, and scalars. This provides the basis for our implementation of the preconditioners in 4D. Extensive numerical experiments illustrate their performance, practical scalability, and parameter robustness, all in accordance with the theory.

math.NA↗

arXiv2017

DPG* Method

We introduce a cousin of the DPG method - the DPG* method - discuss their relationship and compare the two methods through numerical experiments.

math.NA↗

arXiv2017

A spacetime DPG method for the Schrodinger equation

A spacetime Discontinuous Petrov Galerkin (DPG) method for the linear time-dependent Schrodinger equation is proposed. The spacetime approach is particularly attractive for capturing irregular solutions. Motivated by the fact that some irregular Schrodinger solutions cannot be solutions of certain first order reformulations, the proposed spacetime method uses the second order Schrodinger operator. Two variational formulations are proved to be well-posed: a strong formulation (with no relaxation of the original equation) and a weak formulation (also called the ultraweak formulation, that transfers all derivatives onto test functions). The convergence of the DPG method based on the ultraweak formulation is investigated using an interpolation operator. A standalone appendix analyzes the ultraweak formulation for general differential operators. Reports of numerical experiments motivated by pulse propagation in dispersive optical fibers are also included.

math.NA↗

arXiv2016

A scalable preconditioner for a DPG method

We show how a scalable preconditioner for the primal discontinuous Petrov-Galerkin (DPG) method can be developed using existing algebraic multigrid (AMG) preconditioning techniques. The stability of the DPG method gives a norm equivalence which allows us to exploit existing AMG algorithms and software. We show how these algebraic preconditioners can be applied directly to a Schur complement system of interface unknowns arising from the DPG method. To the best of our knowledge, this is the first massively scalable algebraic preconditioner for DPG problems.

math.NA↗

arXiv2015

A tent pitching scheme motivated by Friedrichs theory

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.

math.NA↗

arXiv2014

Five lectures on DPG methods

This is a set of lecture notes introducing graduate students to the topic of Discontinuous Petrov-Galerkin (DPG) methods.

math.NA↗