Generalized Gaffney inequality and discrete compactness for discrete differential forms
This work provides theoretical foundations for analyzing finite element methods in computational electromagnetism and related fields, extending previous results to a broader class of discrete differential forms.
The paper proves generalized Gaffney inequalities and discrete compactness for finite element differential forms on s-regular domains, including Lipschitz domains, without requiring divergence-free constraints. As an application, it provides L^p estimates for finite element approximations of scalar and vector Laplacian problems.
We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on $s$-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have been established for edge elements with weakly imposed divergence-free conditions and used in the analysis of nonlinear and eigenvalue problems. In this paper, we generalize these results to discrete differential forms, not necessarily with strongly or weakly imposed constraints. The analysis relies on a new Hodge mapping and its approximation property. As an application, we show $L^{p}$ estimates for several finite element approximations of the scalar and vector Laplacian problems.