Korn's inequalities for piecewise H1 vector fields

By Susanne C. Brenner. Mathematics of Computation, 2003

357 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/15b62933fc27588dbe52a23a892829e2b507a912

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In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Stabilized Crouzeix‐Raviart element for the Darcy‐Stokes problem (2005, methodology): “In [4] it was shown that J (u, v) controls the rigid body rotations which cause a lack of coercivity for the Crouzeix-Raviart approximation when applied to the Stokes problem written in terms of the symmetric part of the velocity gradient (cf.”
A coupling of nonconforming and mixed finite element methods for Biot's consolidation model (2013, methodology): “As this nonconforming finite element space does not satisfy the classical Korn’s inequality [31, 32], we add a jump (penalty) term to the discrete variational formulation, so that Korn’s inequality for piecewise H 1 vector fields is satisfied [33].”
Finite element methods in linear poroelasticity: theoretical and computational results (2005, methodology): “One such basis is the non-conforming method on triangles, where continuity at the midpoint is demanded [Brenner and Scott, 1994]. However, one possible drawback of this method is its restriction to triangles. In order to disscover another method which is applicable on both triangles and quadrilatera”
Poincaré–Friedrichs Inequalities for Piecewise H 2 Functions (2004, background): “Note that generalized Poincaré–Friedrichs inequalities for piecewise H1 functions and generalized (Korn, inequalities for piecewise H1 vector fields have been established in Brenner 2003, 2004).”
Compact embeddings of broken Sobolev spaces and applications (2009, methodology): “We define broken Sobolev norms for Sobolev indices p ∈ [1,∞) and prove several embedding theorems such as broken Poincaré–Sobolev inequalities (see also [14, 5, 6]) and trace theorems; see Section 4.”
Discontinuous Galerkin methods for solving the Signorini problem (2011, background): “2) is equivalent to the usual DG-norm (| · |(2)1,h + | · |(2)∗), thanks to Korn’s inequality (see [7]; see also Proposition 4.”
DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition (2009, background): “The following Korn’s inequality follows immediately from a result in [8], again established in a more general setting.”
Weak Galerkin method for the coupled Darcy-Stokes flow (2014, background): “By [4], when all K ∈ T S h are affine homeomorphic”
Unified finite element discretizations of coupled Darcy–Stokes flow (2009, background): “Therefore, all the Korn inequalities derived in [6], in particular (3.”
Locking-free Reissner-Mindlin elements without reduced integration (2007, background): “Proofs of the two following lemmata, giving discrete Korn’s inequality and a coercivity estimate, can be found in [ 9 ] and [4].”
A discrete de Rham complex with enhanced smoothness (2006, positive): “From the observation of [11], based on the general nonconforming theory of [3], it follows that the elements of the nonconforming H 1 space Vh indeed satisfy Korn’s inequality.”