Finite Element Systems for Vector Bundles: Elasticity and Curvature

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-022-09555-x Finite Element Systems for Vector Bundles: Elasticity and Curvature Snorre H. Christiansen1 · Kaibo Hu2 Received: 1 April 2020 / Revised: 12 October 2021 / Accepted: 12 October 2021 © The Author(s) 2022 Abstract We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes. Keywords Finite elements · Elasticity · Bianchi identity · de Rham theorem Mathematics Subject Classification 65N30 · 58A10 · 74B05 Introduction In this paper, we first generalize the previously introduced framework of finite element systems (FES) [26,35] so that it can treat, in particular, elasticity problems, and then Communicated by Hans Munthe – Kaas. B Snorre H. Christiansen Kaibo Hu 1 Department of Mathematics, University of Oslo, PO Box 1053, Blindern, 0316 Oslo, Norway 2 Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK 123 Foundations of Computational Mathematics provide concrete examples of finite element spaces, some old and some new, that fit the framework. The general framework provides an approach to finite element discretizations of sections of vector bundles, and complexes thereof, in particular differential forms with values in a given vector bundle. We make some comments about curvature, but most of the paper concerns the case of flat bundles. For applications in elasticity, the fiber can be identified as the space rigid motions. In space dimension 2, one can distinguish between two differential complexes related to elasticity, which are formal adjoints of each other and give priority to stresses and strains, respectively. For the stress complex (62), we can check that variants of the spaces defined in [55] and [4] fit the framework. For the strain complex (64), we introduce, also within the framework, some new finite element spaces. They model symmetric 2-tensors (metrics) with a good Saint-Venant operator (linearized curvature). In the obtained finite element complexes, rigid-motion-like degrees of freedom play a key role, at every index. The FES framework stresses this design principle and relates it to the interpretation of elasticity in terms of rigid motion valued fields. Defining discrete spaces of metrics with good curvature in dimension 2 should be useful, in view of the importance of curved surfaces in several branches of mathematics, both pure and applied, whatever the distinction is. Such applications will be explored elsewhere. Another motivation for this work was to prepare the way for similar constructions in higher dimensions, especially 3 (with classical elasticity in mind) and 4 (with general relativity in mind). Previous work on FES Until now, the FES framework has been formulated in order to discretize de Rham complexes. It has been used to define finite element complexes of differential forms on polyhedral meshes [26], accommodate upwinded finite element complexes containing exponentials [28,33], give new presentations of known elements [38] and to define elements with minimal dimension [30] under various constraints (such as containing given polynomials). The regularity of the differential forms, in the above-mentioned works, was L2 with exterior derivative in L2 , and the defined finite elements were natural generalizations of, in particular, the Raviart–Thomas–Nédélec (RTN) spaces [63,70]. The continuity is thus partial, and can be expressed as singlevaluedness of pullbacks to interfaces, corresponding, for vector fields, to continuity in either tangential or normal directions. In [35], we extended the FES framework so as to be able to impose stronger interelement continuity. For instance, for a conforming discretization of the Stokes equation, one would like to have spaces of fully continuous vector fields, satisfying a commuting diagram with respect to the divergence operator. For de Rham sequences of higher regularity (H1 and, if desired, exterior derivative in H1 ), the required continuity can be expressed as singlevaluedness of all components of the differential form and, if desired, of its exterior derivative too, on interfaces. This led us, in [35], to define FE complexes starting with the Clough–Tocher element, which is of class C1 , instead of, say, Lagrange elements, which are of class C0 . This provided the first conforming polynomial composite Stokes element in dimension 3 (and higher), with piecewise 123 Foundations of Computational Mathematics constant divergence and the degrees of freedom of [16]. The latter seem to be the natural ones for lowest order approximations. FE, MFE, FEEC, VEM Recall Ciarlet’s definition of a finite element (FE), as a space equipped with degrees of freedom [41]. For mixed finite element methods (MFE), pairs of finite element spaces that are compatible in the sense of Brezzi [21] should be identified. A particularly convenient tool for this purpose has been the so-called commuting diagram property, see for instance [72] page 552 and 570 and compare with [17] §8.4 and §8.5. It can sometimes be derived from a commutation property of the interpolators associated with the degrees of freedom. In particular, in [64], finite element grad − curl − div complexes were presented with degrees of freedom providing commuting diagrams. Arbitrary order finite element complexes of differential forms were defined in [52]. Whitney forms [80,81] and the RTN spaces appear as special cases (lowest order—arbitrary dimension, and arbitrary order—low dimension, respectively). This connection between numerical methods and differential topology was first pointed out in [19]. Computational electromagnetics has been one of the main motivations [20,53,65]. Its interpretation in terms of differential forms is quite clearcut compared with the case for, say, computational fluid dynamics. Systematically developing the theory of finite elements in terms of differential complexes equipped with commuting projections was advocated in [2]. Relating de Rham complexes to differential complexes appearing in elasticity, and viewing both as special cases of complexes of Hilbert spaces, has lead to the finite element exterior calculus (FEEC) [3,7,9]. Stability of numerical methods is, in many cases, equivalent t (...truncated)