Uniform Poincaré inequalities for the discrete de Rham complex of differential forms
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In this talk, we prove discrete Poincaré inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed by Bonaldi, Di Pietro, Droniou, and Hu in [1]. We unify the ideas underlying Poincaré inequalities for all differential operators in the sequence, extending existing results for the gradient, curl, and divergence to polyhedral domains of arbitrary dimension and general topology. A key step in the proof is deriving specific Poincaré inequalities for the cochain complex associated with the polyhedral mesh. These inequalities are of independent interest, as they are useful, for instance, in establishing the existence and stability of solutions for mimetic numerical schemes.