POINCAR´ E-FRIEDRICHS INEQUALITIES FOR PIECEWISE H 1 FUNCTIONS ∗

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Conforming and nonconforming virtual element methods for elliptic problems (2015, unknown): “It may be seen that the broken Sobolev norm ) and hence the same will be true for any virtual element subspace V h , as required by Assumption A1, cf. [12].”
Compact embeddings of broken Sobolev spaces and applications (2009, methodology): “We note also that in dimension= 2, 3, such a submesh can be constructed under fairly mild assumptions on the partitionTh (Brenner, 2003, Corollary 7.3).”
Forty Years of the Crouzeix‐Raviart element (2015, methodology): “Using enriching operators one can also establish Poincaré-Friedrichs inequalities for piecewise H 1 and H 2 functions [37,38], and Korn’s inequalities for piecewise H 2 vector fields [39].”
The nonconforming Virtual Element Method for eigenvalue problems (2018, methodology): “Moreover, using the Poincar´e-Friedrichs inequality || v || 20 ≤ C PF | v | 21 ,h , which holds for every v ∈ H 1 ,nc (Ω h ; k ) and some positive constant C PF independent of h , cf. [20], we can show that | · | 1 ,h is equivalent to || · || 1 ,h .”
The nonconforming virtual element method (2014, methodology): “For an integer k ≥ 1, we deﬁne In fact, a standard application of the results in [12] shows that a Poincare inequality holds for functions in H 1 ,nc ( T h ) (already with k = 1), i.e, there exists a constant C P > 0 independent of h such that Therefore with a small abuse of notation we will refer…”