A Spacetime DPG Method for the Schrödinger Equation

Demkowicz, Leszek; Gopalakrishnan, Jay; Nagaraj, Sriram; Sepulveda, Paulina

doi:10.1137/16m1099765

Cited by 48 publications

(46 citation statements)

References 22 publications

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“…The first statement is exactly the statement of [23, Theorem A.5]. The second statement also follows from [23,Theorem A.5] when L is replaced by L * .…”

Section: Ultraweak Formulationssupporting

confidence: 57%

The DPG-star method

Demkowicz

Gopalakrishnan

Keith

2020

Computers & Mathematics with Applications

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A b s t r ac t . 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 LL * 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.

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“…The first statement is exactly the statement of [23, Theorem A.5]. The second statement also follows from [23,Theorem A.5] when L is replaced by L * .…”

Section: Ultraweak Formulationssupporting

confidence: 57%

The DPG-star method

Demkowicz

Gopalakrishnan

Keith

2020

Computers & Mathematics with Applications

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“…Its definition is arbitrary, but fixed once and for all. To avoid the proliferation of constants, the reference length Ω is chosen such that (15) ||β||…”

Section: Standard Sobolev Spaces and Péclet Numbermentioning

confidence: 99%

“…In recent years, the discontinuous Petrov-Galerkin method with optimal test functions ("DPG method" in the following) has proved to be an attractive strategy to produce infsup stable approximations for a wide class of problems. The basic setting stems from Demkowicz and Gopalakrishnan [14,13] and has been extended, e.g., to linear elasticity [1,18], the Stokes and Maxwell equations [28,7], the Schrödinger equation [15], boundary integral and fractional equations [24,17]. Another promising application area is singularly perturbed problems [16,9,3,4,25].…”

mentioning

confidence: 99%

A DPG Framework for Strongly Monotone Operators

Cantin

Heuer

2018

SIAM J. Numer. Anal.

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Abstract. We present and analyze a hybrid technique to numerically solve strongly monotone nonlinear problems by the discontinuous Petrov-Galerkin method with optimal test functions (DPG). Our strategy is to relax the nonlinear problem to a linear one with additional unknown and to consider the nonlinear relation as a constraint. We propose to use optimal test functions only for the linear problem and to enforce the nonlinear constraint by penalization. In fact, our scheme can be seen as a minimum residual method with nonlinear penalty term. We develop an abstract framework of the relaxed DPG scheme and prove under appropriate assumptions the well-posedness of the continuous formulation and the quasi-optimal convergence of its discretization. As an application we consider an advection-diffusion problem with nonlinear diffusion of strongly monotone type. Some numerical results in the lowest-order setting are presented to illustrate the predicted convergence.Key words. Discontinuous Petrov-Galerkin method, optimal test functions, strongly monotone operator, advectiondiffusion, nonlinear penalty.AMS subject classifications. 65N30, 65J15, 65N12, 47H051. Introduction. In recent years, the discontinuous Petrov-Galerkin method with optimal test functions ("DPG method" in the following) has proved to be an attractive strategy to produce infsup stable approximations for a wide class of problems. The basic setting stems from Demkowicz and Gopalakrishnan [14,13] and has been extended, e.g., to linear elasticity [1,18], the Stokes and Maxwell equations [28,7], the Schrödinger equation [15], boundary integral and fractional equations [24,17]. Another promising application area is singularly perturbed problems [16,9,3,4,25].All the cited references, however, deal with linear problems. An extension of the DPG technology to nonlinear problems, on the other hand, is a delicate issue. Principal problem is that the calculation (or approximation) of optimal test functions involves an application of the underlying operator (the DPG method is a minimum residual method). For nonlinear problems this step thus becomes nonlinear, i.e., expensive. One way to circumvent the nonlinearity is, of course, to linearize the underlying problem. This has been the approach in [8,29]. A different idea is to apply the minimum residual technique in product or "broken" spaces to the nonlinear problem. Bui-Thanh and Ghattas [5] did this by considering the entire nonlinear problem as a constraint, and Carstensen et al.[6] developed a representation of the DPG scheme by a nonlinear mixed form and analyzed the case of lowest order approximations. DPG for contact problems has been studied in [21], though in this case the nonlinearity is due to the contact condition which is treated by a variational inequality. We also note that Muga and van der Zee [27] study problems posed in Banach spaces. In those cases the calculation of optimal test functions becomes nonlinear even though the underlying PDE is linear.In this paper we propose a combined scheme that empl...

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“…In choosing the best algorithm to solve the least-squares problem coming from a DLS method, many factors are important to consider. For instance, the normal equation have been demonstrated to be adequate when the methodology has been applied to many DPG problems [64,65,21,67,33,37,49,62,30,40,38,35]. Indeed, in many reasonable circumstances, the round-off error in the solution from the associated linear solve cannot be expected to be nearly as large as the truncation error due to the finite element discretization.…”

Section: Solution Algorithmsmentioning

confidence: 99%

“…Because G can be efficiently inverted with DPG, in that setting, much smaller problems, posed solely in the primal variable u, can be solved directly. This has been performed for many problems of engineering interest [65,21,67,33,37,49,62,30,40,38,35].…”

Section: Introductionmentioning

confidence: 99%

Discrete least-squares finite element methods

Keith

Petrides

Fuentes

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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ABSTRACT. A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number O(h −1 ) for a variety of formulations of Poisson's equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson's equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson's equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion. The methodology described here was developed as an outgrowth of the discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [29]. The strength of DPG is highlighted throughout, however, the majority of theory presented is general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

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A Spacetime DPG Method for the Schrödinger Equation

Cited by 48 publications

References 22 publications

The DPG-star method

The DPG-star method

A DPG Framework for Strongly Monotone Operators

Discrete least-squares finite element methods

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