Stabilized Finite Elements for a Reaction-Dispersion Saddle-Point Problem with NonConstant Coefficients

Belgacem, Faker Ben; Bernardi, Christine; Hecht, Frédéric; Salmon, Stéphanie

doi:10.1137/130907240

Cited by 2 publications

(21 citation statements)

References 14 publications

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“…We recall that this situation is already investigated in , Section 4] when the physical parameters are constant but seem close to real‐life values.…”

Section: Numerical Experimentsmentioning

confidence: 54%

“…We recall that, according to the analysis of the abstract problem achieved in (see also ), necessary and sufficient conditions on the bilinear forms are required to ensure existence, uniqueness, and stability for the mixed problem . The inf‐sup conditions on the forms m (·,·) and m ∗ (·,·) easily follow from the imbedding of

H_{0}^{1} (normalΩ)

into

double-struckV

, see , Lemmas and 2.3]. Lemma The bilinear form m (·,·) satisfies the inf‐sup condition for a positive constant η

\forall ψ \in H_{0}^{1} (normalΩ), \sup_{φ \in double-struckV} \frac{m (ψ, φ)}{∥ φ ∥_{double-struckV}} \geq η ∥ ψ ∥_{H^{1} (normalΩ)} .

The bilinear form m ∗ (·,·) satisfies the inf‐sup condition for a positive constant η ∗

\forall ψ \in H_{0}^{1} (normalΩ), \sup_{φ \in double-struckV} \frac{m_{*} (ψ, φ)}{∥ φ ∥_{double-struckV}} \geq η_{*} ∥ ψ ∥_{H^{1} (normalΩ)} .

…”

Section: The Mixed Variational Frameworkmentioning

confidence: 99%

“…The inf‐sup conditions on a (·,·) use the kernels of the bilinear forms m (·,·) and m ∗ (·,·) defined to be

\begin{matrix} scriptK & = {φ \in double-struckV; \forall ψ \in H_{0}^{1} (normalΩ), m (ψ, φ) = 0}, \\ {scriptK}_{*} & = {φ \in double-struckV; \forall ψ \in H_{0}^{1} (normalΩ), m_{*} (ψ, φ) = 0} . \end{matrix}

Note that

scriptK

and

{scriptK}_{*}

are closed subspaces in

double-struckV

, and hence Hilbert spaces when endowed with

∥ \cdot ∥_{double-struckV}

. We now recall the inf‐sup conditions on the form a (·,·) from , Lemma 2.6], their proof relies on the construction of an isomorphism between the spaces

scriptK

and

{scriptK}_{*}

. Lemma Asume that

osc \sqrt{\frac{r_{*}}{r}} = \max_{bold-italicx \in \overset{normalΩ}{false}} \sqrt{\frac{r_{*}}{r} (bold-italicx)} - \min_{bold-italicx \in \overset{normalΩ}{false}} \sqrt{\frac{r_{*}}{r} (bold-italicx)} < 2 .

Then, the bilinear form a (·,·) satisfies the two inf‐sup conditions for a positive constant τ

\begin{matrix} \forall χ \in \end{matrix}

…”

Section: The Mixed Variational Frameworkmentioning

confidence: 99%

“…We are now in a position to state and prove the next lemma. Its rather technical proof relies on the same arguments as for , Lemma 2.6]. Lemma Assume that condition holds.…”

Section: The Discrete Problem and Its Well‐posednessmentioning

confidence: 99%

“…Figure shows the approximation of the ellipse by the union of the parallelipedic domains. As proposed in , Section 5], the dispersion and reaction parameters are constant and fixed to

d = 0.15, r = 0.2, r_{*} = 0.4 .

…”

Section: Numerical Experimentsmentioning

confidence: 99%

See 4 more Smart Citations

Spectral discretization of a model for organic pollution in waters

Bernardi

Yakoubi

2016

Math Methods in App Sciences

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We are interested in a mixed reaction diffusion system describing the organic pollution in stream‐waters. In this work, we propose a mixed‐variational formulation and recall its well‐posedness. Next, we consider a spectral discretization of this problem and establish nearly optimal error estimates. Numerical experiments confirm the interest of this approach. Copyright © 2016 John Wiley & Sons, Ltd.

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“…We recall that this situation is already investigated in , Section 4] when the physical parameters are constant but seem close to real‐life values.…”

Section: Numerical Experimentsmentioning

confidence: 54%

H_{0}^{1} (normalΩ)

into

double-struckV

, see , Lemmas and 2.3]. Lemma The bilinear form m (·,·) satisfies the inf‐sup condition for a positive constant η

\forall ψ \in H_{0}^{1} (normalΩ), \sup_{φ \in double-struckV} \frac{m (ψ, φ)}{∥ φ ∥_{double-struckV}} \geq η ∥ ψ ∥_{H^{1} (normalΩ)} .

The bilinear form m ∗ (·,·) satisfies the inf‐sup condition for a positive constant η ∗

\forall ψ \in H_{0}^{1} (normalΩ), \sup_{φ \in double-struckV} \frac{m_{*} (ψ, φ)}{∥ φ ∥_{double-struckV}} \geq η_{*} ∥ ψ ∥_{H^{1} (normalΩ)} .

…”

Section: The Mixed Variational Frameworkmentioning

confidence: 99%

“…The inf‐sup conditions on a (·,·) use the kernels of the bilinear forms m (·,·) and m ∗ (·,·) defined to be

\begin{matrix} scriptK & = {φ \in double-struckV; \forall ψ \in H_{0}^{1} (normalΩ), m (ψ, φ) = 0}, \\ {scriptK}_{*} & = {φ \in double-struckV; \forall ψ \in H_{0}^{1} (normalΩ), m_{*} (ψ, φ) = 0} . \end{matrix}

Note that

scriptK

and

{scriptK}_{*}

are closed subspaces in

double-struckV

, and hence Hilbert spaces when endowed with

∥ \cdot ∥_{double-struckV}

. We now recall the inf‐sup conditions on the form a (·,·) from , Lemma 2.6], their proof relies on the construction of an isomorphism between the spaces

scriptK

and

{scriptK}_{*}

. Lemma Asume that

osc \sqrt{\frac{r_{*}}{r}} = \max_{bold-italicx \in \overset{normalΩ}{false}} \sqrt{\frac{r_{*}}{r} (bold-italicx)} - \min_{bold-italicx \in \overset{normalΩ}{false}} \sqrt{\frac{r_{*}}{r} (bold-italicx)} < 2 .

Then, the bilinear form a (·,·) satisfies the two inf‐sup conditions for a positive constant τ

\begin{matrix} \forall χ \in \end{matrix}

…”

Section: The Mixed Variational Frameworkmentioning

confidence: 99%

“…We are now in a position to state and prove the next lemma. Its rather technical proof relies on the same arguments as for , Lemma 2.6]. Lemma Assume that condition holds.…”

Section: The Discrete Problem and Its Well‐posednessmentioning

confidence: 99%

“…Figure shows the approximation of the ellipse by the union of the parallelipedic domains. As proposed in , Section 5], the dispersion and reaction parameters are constant and fixed to

d = 0.15, r = 0.2, r_{*} = 0.4 .

…”

Section: Numerical Experimentsmentioning

confidence: 99%

See 3 more Smart Citations

Spectral discretization of a model for organic pollution in waters

Bernardi

Yakoubi

2016

Math Methods in App Sciences

Self Cite

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An ill-posed parabolic evolution system for dispersive deoxygenation–reaeration in water

et al. 2013

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We consider an inverse problem that arises in the management of water resources and pertains to the analysis of the surface waters pollution by organic matter. Most of physical models used by engineers derive from various additions and corrections to enhance the earlier deoxygenationreaeration model proposed by Streeter and Phelps in 1925, the unknowns being the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) concentrations. The one we deal with includes Taylor's dispersion to account for the heterogeneity of the contamination in all space directions. The system we obtain is then composed of two reaction-dispersion equations. The particularity is that both Neumann and Dirichlet boundary conditions are available on the DO tracer while the BOD density is free of any condition. In fact, for real-life concerns, measurements on the dissolved oxygen are easy to obtain and to save. In the contrary, collecting data on the biochemical oxygen demand is a sensitive task and turns out to be a long-time process. The global model pursues the reconstruction of the BOD density, and especially of its flux along the boundary. Not only this problem is plainly worth studying for its own interest but it can be also a mandatory step in other applications such as the identification of the pollution sources location. The non-standard boundary conditions generate two difficulties in mathematical and computational grounds. They set up a severe coupling between both equations and they are cause of ill-posedness for the data reconstruction problem. Existence and stability fail. Identifiability is therefore the only positive result one can seek after ; it is the central purpose of the paper. We end by some computational experiences to assess the capability of the mixed finite element capability in the missing data recovery (on the biochemical oxygen demand).

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Stabilized Finite Elements for a Reaction-Dispersion Saddle-Point Problem with NonConstant Coefficients

Cited by 2 publications

References 14 publications

Spectral discretization of a model for organic pollution in waters

Spectral discretization of a model for organic pollution in waters

An ill-posed parabolic evolution system for dispersive deoxygenation–reaeration in water

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