Adaptivity and variational stabilization for convection-diffusion equations

Cohen, Albert; Dahmen, Wolfgang; Welper, Gerrit

doi:10.1051/m2an/2012003

Cited by 114 publications

(108 citation statements)

References 22 publications

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“…Second, it suggests an "ideal setting" for Petrov-Galerkin schemes from which numerically feasible versions can be derived in a systematic manner. A slightly different but related approach has been proposed in [9] for a more restricted scope of problems.…”

Section: Abstract Frameworkmentioning

confidence: 99%

Adaptive Petrov--Galerkin Methods for First Order Transport Equations

Dahmen¹,

Huang²,

Schwab³

et al. 2012

SIAM J. Numer. Anal.

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106

171

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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.

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Section: Abstract Frameworkmentioning

confidence: 99%

Adaptive Petrov--Galerkin Methods for First Order Transport Equations

Dahmen¹,

Huang²,

Schwab³

et al. 2012

SIAM J. Numer. Anal.

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106

171

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“…In fact, in the linear case it is well known that there is a mixed form of the DPG scheme, and this is precisely the method proposed (for a specific model problem) by Cohen et al [11]. As we will see, our scheme can also be viewed as an extension of this mixed form.…”

mentioning

confidence: 71%

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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“…Introducing (11) in the stationary condition associated with problem (10), one can derive a gradient-type algorithm that can be viewed as an extension of algorithms introduced in [17,14] to nonlinear approximation in S M . This algorithm reads at a given iteration ðk þ 1Þ: knowing u…”

Section: Direct Construction With Low-rank Approximation Of An Auxilimentioning

confidence: 99%

Ideal minimal residual-based proper generalized decomposition for non-symmetric multi-field models – Application to transient elastodynamics in space-time domain

Boucinha

Ammar

Gravouil

et al. 2014

Computer Methods in Applied Mechanics and Engineering

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. b s t r a c tIt is now well established that separated representations built with the help of proper generalized decomposition (PGD) can drastically reduce computational costs associated with solution of a wide variety of problems. However, it is still an open question to know if separated representations can be efficiently used to approximate solutions of hyperbolic evolution problems in space-time domain. In this paper, we numerically address this issue and concentrate on transient elastodynamic models. For such models, the operator associated with the space-time problem is non-symmetric and low-rank approximations are classically computed by minimizing the space-time residual in a natural L 2 sense, yet leading to non optimal approximations in usual solution norms. Therefore, a new algorithm has been recently introduced by one of the authors and allows to find a quasi-optimal low-rank approximation a priori with respect to a target norm. We presently extend this new algorithm to multi-field models. The proposed algorithm is applied to elastodynamics formulated over space-time domain with the Time Discontinuous Galerkin method in displacement and velocity. Numerical examples demonstrate convergence of the proposed algorithm and comparisons are made with classical a posteriori and a priori approaches.

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Adaptivity and variational stabilization for convection-diffusion equations

Cited by 114 publications

References 22 publications

Adaptive Petrov--Galerkin Methods for First Order Transport Equations

Adaptive Petrov--Galerkin Methods for First Order Transport Equations

A DPG Framework for Strongly Monotone Operators

Ideal minimal residual-based proper generalized decomposition for non-symmetric multi-field models – Application to transient elastodynamics in space-time domain

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