Uniform Poincaré inequalities for the discrete de Rham complex on general domains (Q6618297)

In this paper Poincare inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain \(\Omega\) of \(R^3\) are proved. The work is organized as follows. Section 1 is an introduction. The definitions of the relevant DDR spaces and operators are briefly recalled in Section 2. Mimetic Poincares inequalities are derived in Section 3, and then used to prove discrete Poincares inequalities for the DDR complex in Section 4. The latter is used in Section 5 to carry out the stability analysis of a DDR scheme for the magnetostatics problem on domains with general topology. Some arguments in the proofs of mimetic Poincare inequalities rely on specific shape functions for Finite Element spaces on a submesh, whose definitions and properties are summarised in the Appendix.