Broken-FEEC discretizations and Hodge Laplace problems

This article studies structure-preserving discretizations of Hilbert complexes with nonconforming (broken) spaces that rely on projection operators onto an underlying conforming subcomplex. This approach follows the conforming/nonconforming Galerkin (CONGA) method introduced by Campos Pinto and Sonnendrücker [Math. Comp. 85 (2016), pp. 2651–2685; SMAI J. Comput. Math. 3 (2017), pp 53–89; SMAI J. Comput. Math. 3 (2017), pp. 91–116] 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 resulting 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 in standard finite element applications, 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 that of 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 show optimal convergence rates despite the lack of a formal stability result for the associated conforming projections. Applications to spline finite elements on multi-patch mapped domains are described in a related article (see Y. Güçlü, S. Hadjout, and M. Campos Pinto [J. Sci. Comput. 97 (2023)]), for which the present work provides a theoretical background.