  This article studies structure-preserving discretizations of Hilbert
complexes with nonconforming spaces that rely on projections onto an underlying
conforming subcomplex. This approach follows the conforming/nonconforming
Galerkin (CONGA) method introduced in [doi.org/10.1090/mcom/3079,
doi.org/10.5802/smai-jcm.20, doi.org/10.5802/smai-jcm.21] to derive efficient
structure-preserving finite element schemes for the time-dependent Maxwell and
Maxwell-Vlasov systems by relaxing the curl-conforming constraint in finite
element exterior calculus (FEEC) spaces. Here, it is extended to the
discretization of full Hilbert complexes with possibly nontrivial harmonic
fields, and the properties of the CONGA Hodge Laplacian operator are
investigated. By using block-diagonal mass matrices which may be locally
inverted, this framework possesses a canonical sequence of dual commuting
projection operators which are local, and it naturally yields local discrete
coderivative operators, in contrast to conforming FEEC discretizations. The
resulting CONGA Hodge Laplacian operator is also local, and its kernel consists
of the same discrete harmonic fields as the underlying conforming operator,
provided that a symmetric stabilization term is added to handle the space
nonconformities. Under the assumption that the underlying conforming subcomplex
admits a bounded cochain projection, and that the conforming projections are
stable with moment-preserving properties, a priori convergence results are
established for both the CONGA Hodge Laplace source and eigenvalue problems.
Our theory is finally illustrated with a spectral element method, and numerical
experiments are performed which corroborate our results. Applications to spline
finite elements on multi-patch mapped domains are described in a related
article [arXiv:2208.05238] for which the present work provides a theoretical
background.
