A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation

Demkowicz, Leszek; Gopalakrishnan, Jay

doi:10.1016/j.cma.2010.01.003

Cited by 225 publications

(260 citation statements)

References 17 publications

Supporting

Mentioning

260

Contrasting

Order By: Relevance

“…After completion of this work we became aware of the work in [14] which is related to the present approach in that it also starts from similar principles of choosing an ideal variational setting. It seems to pursue though a different direction by computing "near-optimal" test functions, exploiting as much as possible the localization offered by a Discontinuous Galerkin context.…”

Section: 1)mentioning

confidence: 99%

“…Alternatively, other local enrichment strategies such as increasing the polynomial order or augmentation by so-called "bubble functions" are conceivable, see e.g. [14,2].…”

Section: 2mentioning

confidence: 99%

“…Of course, employing the ideal test space [14] is to approximate individually ideal test basis functions. Since stability of the PG discretization depends on subspaces rather than individual basis elements, it is more appropriate to approximate whole subspaces.…”

Section: Perturbing the Ideal Casementioning

confidence: 99%

See 2 more Smart Citations

Adaptive Petrov--Galerkin Methods for First Order Transport Equations

Dahmen¹,

Huang²,

Schwab³

et al. 2012

SIAM J. Numer. Anal.

106

171

View full text Add to dashboard Cite

Abstract. We propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport and evolution equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value problems for these operators. To adaptively resolve anisotropic solution features such as propagating singularities the variational formulations should allow one, in particular, to employ as trial spaces directional representation systems. Since such systems are known to be stable in L 2 special emphasis is placed on L 2 -stable formulations. The proposed stability concept is based on perturbations of certain "ideal" test spaces in Petrov-Galerkin formulations. We develop a general strategy for realizing corresponding schemes without actually computing excessively expensive test basis functions. Moreover, we develop adaptive solution concepts with provable error reduction. The results are illustrated by first numerical experiments.

show abstract

Section: 1)mentioning

confidence: 99%

“…Alternatively, other local enrichment strategies such as increasing the polynomial order or augmentation by so-called "bubble functions" are conceivable, see e.g. [14,2].…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Adaptive Petrov--Galerkin Methods for First Order Transport Equations

Dahmen¹,

Huang²,

Schwab³

et al. 2012

SIAM J. Numer. Anal.

106

171

View full text Add to dashboard Cite

show abstract

“…However, one is then left with the question of how to choose the test space. Recent works on optimal or near-optimal choice of Petrov-Galerkin test spaces were presented in [40,41] and [38]. These options are "optimal" in the sense that they give the best possible ratio of continuity constant to stability constant in the energy norm estimates.…”

Section: Residualsmentioning

confidence: 99%

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

161

179

View full text Add to dashboard Cite

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

show abstract

“…These methods have been extensively employed in the last years in applied mathematics to solve a variety of engineering problems. For example, the isogeometric analysis (IGA) [8,9] has recently experimented a huge explosion and it has been widely applied to the engineering industry, as well as the more recent Discontinuous Petrov-Galerkin (DPG) method initially proposed by Demkowicz and Gopalakrishnan [10,11], or the self-adaptive hp-Finite Element Method (FEM) [12,13] (where h stands for the element size and p for the polynomial order of approximation associated to each element). The latter one has been recently employed, for instance, to model the bone conduction of sound in the human head [14], or to simulate bend, step, and magic-T electromagnetic waveguide structures [15].…”

Section: Introductionmentioning

confidence: 99%