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.

翻译：本文研究基于投影至底层协调子复形的非协调空间的希尔伯特复形保持结构离散化。该方法遵循文献[doi.org/10.1090/mcom/3079, doi.org/10.5802/smai-jcm.20, doi.org/10.5802/smai-jcm.21]中提出的协调/非协调伽辽金(CONGA)方法，通过松弛有限元外微积分(FEEC)空间中的旋度协调约束，推导时变麦克斯韦和麦克斯韦-弗拉索夫系统的高效保结构有限元格式。本文将其扩展至可能包含非平凡调和场的完整希尔伯特复形离散化，并研究CONGA霍奇拉普拉斯算子的性质。通过使用可局部求逆的分块对角质量矩阵，该框架拥有典范的局部对偶交换投影算子序列，并自然产生局部离散余微分算子——这与协调FEEC离散化形成对比。由此产生的CONGA霍奇拉普拉斯算子亦为局部的，且其核空间与底层协调算子的核空间包含相同的离散调和场，前提是添加对称稳定项以处理空间非协调性。在假设底层协调子复形存在有界上链投影，且协调投影具有保持矩性质的稳定性条件下，建立了CONGA霍奇拉普拉斯源问题和特征值问题的先验收敛结果。最终以谱元法为例说明理论，并通过数值实验验证结论。关于多拼贴映射域上样条有限元的应用已在相关论文[arXiv:2208.05238]中描述，本文为其提供理论基础。