SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs

Breit, Dominic; Diening, Lars; Schwarzacher, Sebastian

doi:10.1142/s0218202513500437

Cited by 57 publications

(92 citation statements)

References 12 publications

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“…This discrete Lipschitz truncation is a composition of a continuous Lipschitz truncation with a projection onto the finite element space. The continuous Lipschitz truncation used here is based on results from [DMS08,BDF12,BDS13], which provides finer estimates than the original Lipschitz truncation technique proposed by Acerbi and Fusco in [AF88]; for details consider [DKS13].…”

Section: We Have That There Exists a Not Relabelled Weakly Convergingmentioning

confidence: 99%

Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

Kreuzer

Süli

2016

ESAIM: M2AN

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Abstract. We develop the a posteriori error analysis of finite element approximations to implicit power-law-like models for viscous incompressible fluids in d space dimensions, d P t2, 3u. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi-valued, maximal monotone r-graph, with 2d d`1 ă r ă 8. We establish upper and lower bounds on the finite element residual, as well as the local stability of the error bound. We then consider an adaptive finite element approximation of the problem, and, under suitable assumptions, we show the weak convergence of the adaptive algorithm to a weak solution of the boundary-value problem. The argument is based on a variety of weak compactness techniques, including Chacon's biting lemma and a finite element counterpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions, introduced by L. Diening, C. Kreuzer and E. Süli [Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal., 51(2), 984-1015].

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Section: We Have That There Exists a Not Relabelled Weakly Convergingmentioning

confidence: 99%

Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

Kreuzer

Süli

2016

ESAIM: M2AN

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“…The bound p > • It is not clear if it is possible to improve the result from Theorem 3 to p > 2d d+2 as in the deterministic case. The papers [16] and [7] use the Lipschitz truncation method. Despite the L ∞ -truncation the Lipschitz truncation is not only nonlinear but also nonlocal (in space-time in the parabolic case).…”

Section: Remarkmentioning

confidence: 99%

“…In this situation we have that (∇v)v ∈ L 1 (Q) and therefore we can test with functions from L ∞ (Q). The basic idea (which was already used in the stationary case in [21] together with the bound p ≥ 2d d+1 ) is to approximate v by a bounded function v λ which is equal to v on a large set and its L ∞ -norm can be controlled by λ. Wolf's result was improved to p > 2d d+2 in [16] and [7] by the Lipschitz truncation method. Under this restriction to p we have v ⊗ v ∈ L 1 (Q) which means we can test by Lipschitz continuous functions.…”

Section: Introductionmentioning

confidence: 99%

Existence Theory for Stochastic Power Law Fluids

Breit

2015

J. Math. Fluid Mech.

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abstract:We consider the equations of motion for an incompressible Non-Newtonian fluid in a bounded Lipschitz domain G ⊂ R d during the time interval (0, T ) together with a stochastic perturbation driven by a Brownian motion W. The balance of momentum reads aswhere v is the velocity, π the pressure and f an external volume force. We assume the common power law model S(ε(v)) = 1 + |ε(v)| p−2 ε(v) and show the existence of weak (martingale) solutions pro-Our approach is based on the L ∞ -truncation and a harmonic pressure decomposition which are adapted to the stochastic setting.

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“…Theorem 2.10.43 in [Fed69]) and uses an additional truncation of v with respect to M (v). This can be avoided by proceeding similarly as in [BDF12,BDS13], i.e., extending v |H λ (v) by means of a partition of unity on a Whitney covering of the open and bounded set U λ (v). To this end, we assume w.l.o.g.…”

Section: Discrete Lipschitz Truncationmentioning

confidence: 99%

Finite Element Approximation of Steady Flows of Incompressible Fluids with Implicit Power-Law-Like Rheology

Diening¹,

Kreuzer²,

Süli³

2013

SIAM J. Numer. Anal.

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We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multivalued, maximal monotone r-graph with 1 < r < ∞. Using a variety of weak compactness techniques, including Chacon's biting lemma and Young measures, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the finite element discretization parameter h tends to 0. A key new technical tool in our analysis is a finite element counterpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions.

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SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs

Cited by 57 publications

References 12 publications

Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

Existence Theory for Stochastic Power Law Fluids

Finite Element Approximation of Steady Flows of Incompressible Fluids with Implicit Power-Law-Like Rheology

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