A finite element method for a noncoercive elliptic problem with Neumann boundary conditions

Kavaliou, Klim; Tobiska, Lutz

doi:10.1002/pamm.201210324

Cited by 2 publications

(6 citation statements)

References 6 publications

(6 reference statements)

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“…For a detailed analysis of the well-posedness of the continuous problems we refer to [14,16] and for a finite element analysis in the case of homogeneous Neumann conditions to [10]. Recent work on numerical methods for these problems have focused on finite volume methods, see [15,11] or hybrid finite element/finite volume methods [19].…”

Section: Nonsymmetric Indefinite Elliptic Problemsmentioning

confidence: 99%

Stabilized Finite Element Methods for Nonsymmetric, Noncoercive, and Ill-Posed Problems. Part I: Elliptic Equations

Burman¹

2013

SIAM J. Sci. Comput.

102

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In this paper we propose a new method to stabilize nonsymmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilized finite element method. Both stabilization of the element residual and of the jumps of certain derivatives of the discrete solution over element faces may be used. Under the assumption of well-posedness of the partial differential equation and its associated adjoint problem we prove optimal error estimates in H 1 and L 2 norms in an abstract framework. Some examples of problems that are neither symmetric nor coercive but that enter the abstract framework are given. First we treat indefinite convectiondiffusion equations with nonsolenoidal transport velocity and either pure Dirichlet conditions or pure Neumann conditions and then a Cauchy problem for the Helmholtz operator. Some numerical illustrations are given. Introduction.The computation of indefinite elliptic problems often involves certain conditions on the mesh size h for the system to be well-posed and for the derivation of error estimates. The first results on this problem are due to Schatz [19]. The conditions on the mesh parameter can be avoided if a stabilized finite element method is used. Such methods have been proposed by Bramble, Lazarov, and Pasciak [4] and Ku [16] or more recently the continuous interior penalty (CIP) method for the Helmholtz equation suggested by Wu [21], and Zhu, Burman, and Wu [20]. The method proposed herein has some common features with both these methods but appears to have a wider field of applicability. We may treat not only symmetric indefinite problems such as the (real valued) Helmholtz equation but also nonsymmetric indefinite problems such as convection-diffusion problems with nonsolenoidal convection velocity or the Cauchy problem. The latter problem is known to be illposed in general [1] and will mainly be explored numerically herein. For all these cases we show that if the primal and adjoint problems admit a unique solution with sufficient smoothness the proposed algorithm converges with optimal order. The case of hyperbolic problems is treated in the companion paper [5].The idea of this work is to assume ill-posedness of the discrete form of the PDE and regularize it in the form of an optimization problem under constraints. Indeed we seek to minimize the size of the stabilization operator under the constraint of the discrete variational form. The regularization terms are then chosen from well-known stabilized methods respecting certain design criteria given in an abstract analysis. This leads to an extended method where simultaneously both a primal and a dual

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Section: Nonsymmetric Indefinite Elliptic Problemsmentioning

confidence: 99%

Stabilized Finite Element Methods for Nonsymmetric, Noncoercive, and Ill-Posed Problems. Part I: Elliptic Equations

Burman¹

2013

SIAM J. Sci. Comput.

102

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“…There is, however, an additional subtlety in the very definition of the measure and its approximation. One, basic but crucial, remark is that the solution u to (1)- (11) does not depend on the choice of σ.…”

Section: Main Result: Estimation Of U ´Uhmentioning

confidence: 99%

“…The advection-diffusion equation ( 1) supplied with the data we have just described and the boundary condition (11) can be studied in the context of the Banach-Nečas-Babuška theory. Defining U " V " H 1 0 pΩq, apu, vq " ˆΩ ∇u ¨∇v `pb ¨∇uqv,…”

Section: Inf-sup Theorymentioning

confidence: 99%

“…where U is a Banach space, V is a reflexive Banach space, a P LpU ˆV ; Rq and F P V 1 . The well-posedness of ( 1)- (11), recast as ( 12)-( 13)- (14), is known to be equivalent to the following two conditions:…”

Section: Inf-sup Theorymentioning

confidence: 99%

“…As mentioned above, (2) does not completely characterize σ since a boundary condition and possibly an additional normalization need to be supplied. Because of the homogeneous Dirichlet boundary condition (11) we have imposed on u for (1), it turns out that we have some flexibility on the boundary condition we may impose on σ. In other situations, the integration by parts performed to employ the adjoint equation might require more stringent conditions on σ on the boundary.…”

Section: On the Invariant Measurementioning

confidence: 99%

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Stable approximation of the advection-diffusion equation using the invariant measure

Bris¹,

Legoll²,

Madiot³

2016

Preprint

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We consider an advection-diffusion equation that is both non-coercive and advection-dominated. We present a possible numerical approach, to our best knowledge new, and based on the invariant measure associated to the original equation. The approach has been summarized in [12]. We show that the approach allows for an unconditionally well-posed finite element approximation. We provide a numerical analysis and a set of comprehensive numerical tests showing that the approach can be stable, as accurate as, and more robust than a classical stabilization approach.

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A finite element method for a noncoercive elliptic problem with Neumann boundary conditions

Cited by 2 publications

References 6 publications

Stabilized Finite Element Methods for Nonsymmetric, Noncoercive, and Ill-Posed Problems. Part I: Elliptic Equations

Stabilized Finite Element Methods for Nonsymmetric, Noncoercive, and Ill-Posed Problems. Part I: Elliptic Equations

Stable approximation of the advection-diffusion equation using the invariant measure

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