Vector potentials in three-dimensional non-smooth domains

Amrouche, Chérif; Bernardi, Christine; Dauge, Monique; Girault, Vivette

doi:10.1002/(sici)1099-1476(199806)21:9<823::aid-mma976>3.0.co;2-b

Cited by 783 publications

(1,005 citation statements)

References 20 publications

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“…and applying the incompressibility constraint, we obtain 5) which implies that the pressure can be eliminated from (2.1). Moreover, we also introduce the velocity gradient as an auxiliary variable t := ∇u in Ω. Consequently, we end up with the following system, expressed in terms of the unknowns t, σ, and u:…”

Section: The Boundary Value Problemmentioning

confidence: 99%

A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

Gatica

Ruiz–Baier

Tierra

2016

Computers & Mathematics with Applications

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In this work we develop the a posteriori error analysis of an augmented mixed finite element method for the 2D and 3D versions of the Navier-Stokes equations when the viscosity depends nonlinearly on the module of the velocity gradient. Two different reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions are derived. Our analysis of reliability of the proposed estimators draws mainly upon the global inf-sup condition satisfied by a suitable linearization of the continuous formulation, an application of Helmholtz decomposition, and the local approximation properties of the RaviartThomas and Clément interpolation operators. In addition, differently from previous approaches for augmented mixed formulations, the boundedness of the Clément operator plays now an interesting role in the reliability estimate. On the other hand, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show their efficiency. Finally, several numerical results are provided to illustrate the good performance of the augmented mixed method, to confirm the aforementioned properties of the a posteriori error estimators, and to show the behaviour of the associated adaptive algorithm.

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Section: The Boundary Value Problemmentioning

confidence: 99%

A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

Gatica

Ruiz–Baier

Tierra

2016

Computers & Mathematics with Applications

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“…Since Ω is simply-connected, we recall from [2,Cor. 3.16] that this quantity is a norm, which is equivalent to the graph norm of H(div, Ω) ∩ H(curl, Ω), i.e., that there exists a constant c only depending on Ω such that…”

Section: Analysis Of the Modelmentioning

confidence: 99%

A posteriori error analysis of the time-dependent Stokes equations with mixed boundary conditions

Bernardi

Sayah²

2014

IMA Journal of Numerical Analysis

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Abstract. In this paper we study the time dependent Navier-Stokes problem with mixed boundary conditions. The problem is discretized by the backward Euler's scheme in time and finite elements in space. We establish optimal a posteriori error estimates with two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization. We finish with numerical validation experiments.

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“…Let W be in X(Ω c ) and set W 1 = (1 − χ)W . Then, W 1 is in H 0 (curl; Ω 2 ) ∩ H(div; Ω 2 ) and therefore is in H s (Ω 2 ) (see [1]). …”

Section: Appendixmentioning

confidence: 99%

Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems

Bramble

Pasciak

2006

Math. Comp.

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Abstract. We consider the approximation of the frequency domain threedimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius R t . We also show exponential (in the parameter R t ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer.Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.

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Vector potentials in three-dimensional non-smooth domains

Cited by 783 publications

References 20 publications

A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

A posteriori error analysis of the time-dependent Stokes equations with mixed boundary conditions

Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems

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