Partially adjoint discretizations of adjoint operators and nonconforming finite element exterior calculus

This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finitedimensional operators. A theory of nonconforming finite element exterior calculus (NCFEEC) is developed by constructing a systematic family of nonconforming finite element differential forms with certain structures by the theoretical framework and the formal construction of discretizations; by the NCFEEC, together with the classical FEEC, adjoint properties are reconstructed between discretizations of adjoint operators and between discretizations of variational problems. The adjoint properties concerned in the paper are the closed range theorem and the strong dualities, whose preservations have not been well studied yet. Quantified versions of the closed range theorem are established for both adjoint operators and partially adjoint discretizations. The notion Poincaré-Alexander-Lefschetz (P-A-L for short) type duality is borrowed for operator theory, and horizontal and vertical P-A-L dualities are figured out for adjoint operators and their analogues are established for partially adjoint discretizations. The ABCD, a new kind of discretization method, is motivated by and plays a crucial role in the construction of partially adjoint discretizations. Besides, it can be used for the discretization of single operators; for example, in this paper, de Rham complexes that start with the Crouzeix-Raviart finite element spaces are constructed by series of ABCDs, including both the discretized operators and the domain spaces; commutative diagrams with minimal request of regularities are constructed thereon. The theory of exterior calculus has shown the importance for mathematics and numerics and contains foundational applications of theory of adjoint operators. Adjoint properties are reconstructed by combining the NCFEEC and the classical FEEC, such as the Poincaré-Lefschetz duality preserved as an identity by the partially adjoint discretizations of the adjoint operators, which was not yet obtained before. Further, equivalences are established between the discretizations for the elliptic variational problems and between the discretizations of the Hodge Laplacian problems. This shows how structure-preserving schemes for equations with linear operators can be lead to by partially adjoint discretizations under the new theoretical framework. 2010 Mathematics Subject Classification. Primary 47A05, 47A65, 47N40, 65J10, 65N30.