Poincaré-Friedrichs Inequalities for Piecewise H1 Functions

By Susanne C. Brenner. SIAM Journal on Numerical Analysis, 2003

482 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/517e5726cd1f8db2008b6d77c546003f098ccdff

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Citations in the corpus

Compact embeddings of broken Sobolev spaces and applications (2009, methodology): “Our results are more general than previous ones in that we consider arbitrary Sobolev indices but weaker than those in [5], for example, since we restrict ourselves to a fixed polynomial degree.”
Poincaré-type inequalities for broken Sobolev spaces (2003, methodology): “We shall closely follow the argument presented by Brenner in [4], extending it from L–based norms to L–based norms, 1 ≤ q ≤ 2n/(n− 2), n ≥ 3.”
The Virtual Element Method with curved edges (2017, unknown): “E+ C|v| 1,E. Now(28)followsformtheaboveboundand(25). Letusanalysethebound(29). Firstofall weintroduceausefulscaledPoincaréinequality. Letw∈H1(E) andlet Π0,∂E 0 w:= 1 |∂E| Z ∂E wds (30) thenitholdsthat[13] kw−Π0,∂E 0 wk 0,E≤Ch E|w| 1,E. (31) Letnowev∈H1(E) theharmonicextensionofv∈H1/2(∂E) (cf. Lemma3”
DISCONTINUOUS GALERKIN METHODS FOR THE MULTI-DIMENSIONAL VLASOV–POISSON PROBLEM (2012, unknown): “The first term in (75) is estimated by using standard regularity theorems for the Poisson problem [29] together with the PoincareFriederichs inequality for discrete functions in Qh (see [11]) and estimate (21) from Lemma 2 |(E− e E, Φ− e Φ)|A = ‖E− e E‖0,T x hx ≤ C‖ρ− ρh‖−1,T x hx = sup qh∈Qh R T x ”
Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case (2007, unknown): “We state the following lemma (see [3, 9, 23]).”
DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition (2009, methodology): “We refer to [7] and to Crouzeix [11] for a slightly different regularization.”
Poincaré–Friedrichs Inequalities for Piecewise H 2 Functions (2004, background): “6 of Brenner, 2003, for details) is that under the conditions on i i ∈ I we can construct for each i ∈ I a simplicial triangulation i ∈ i such that i is bounded below by a positive number independent of i ∈ I .”
The nonconforming virtual element method (2014, background): “s= 0 8q2Pk 1(e); 8e2E h ˙ : Although for discontinuous functions jj 1;h is only a semi-norm, for v2V = H1 0 ( ) and v2H1;nc(T h) it is indeed a norm. In fact, a standard application of the results in [12] shows that a Poincare inequality holds for functions in H1;nc(T h) (already with k= 1), i.e, th”
A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity (2009, unknown): “19)), using Vh ⊂ [H(1)(Ω1 ∪ Ω2)] and v|∂Ω = 0 for v ∈ Vh, we have: ‖ε(v)‖20,Ω1∪Ω2 + ‖[v]‖ 2 1/2,h,Γ ≥ C|v|21,Ω1∪Ω2 ∀v ∈ Vh, while from [2], using v|∂Ω = 0, v|ΓK ∈ L(2)(ΓK) for v ∈ Vh and a standard error estimate, we have |v|21,Ω1∪Ω2 + ‖[v]‖ 2 1/2,h,Γ ≥ C‖v‖(2)0,Ω ∀v ∈ Vh and thus ‖ε(v)‖20,Ω1∪Ω2 + ‖”
On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1 (2005, methodology): “S. C. Brenner (2003).”