Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-024-09642-1 Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields Philipp Bringmann1 · Jonas W. Ketteler2 · Mira Schedensack2 Received: 24 April 2023 / Revised: 14 November 2023 / Accepted: 20 November 2023 © The Author(s) 2024 Abstract Discrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well. Keywords Discrete Helmholtz decompositions · Nonconforming FEM · Fortin–Soulie · Crouzeix–Raviart · Mixed FEM · Stokes equations · Linear elasticity · Fourth-order problems Mathematics Subject Classification 65D18 · 65N30 · 74B05 · 74S05 · 76D07 · 76M10 Communicated by Rob Stevenson. B Mira Schedensack Philipp Bringmann Jonas W. Ketteler 1 TU Wien, Institute of Analysis and Scientific Computing, Vienna, Austria 2 Institute of Mathematics, Leipzig University, Leipzig, Germany 123 Foundations of Computational Mathematics 1 Introduction The Helmholtz decomposition describes a vector field on a bounded and contractible domain  ⊆ Rd as the sum of an irrotational and a solenoidal vector field, i.e., L 2 (; Rd ) = ∇ H01 ()  ⊥ rot H (rot, ), (1.1) ⊥ means that the sum is L 2 -orthogonal. It is a fundamental tool for the analwhere  ysis and visualization of vector fields in various areas including fluid mechanics, astrophysics, geophysics, and imaging. For a historical overview of the Helmholtz decomposition on the continuous level, the reader is referred to [53] and [8]. Throughout the paper, let T denote a conforming triangulation of a bounded and polyhedral Lipschitz domain  into closed simplices. This paper investigates discrete versions of the Helmholtz decomposition (1.1) of the form Pk (T ; Rd ) = ∇NC X h (T )  ⊥ rot NC Yh (T ) (1.2) for k = 0, 1 and d = 2, 3. At least one of the discrete spaces X h (T ) and Yh (T ) has to be nonconforming and the differential operators ∇NC and rot NC apply piecewise. Such a decomposition was proved for the first time by Arnold and Falk for k = 0 and d = 2 in [2] with X h (T ) being the Crouzeix–Raviart finite element space and Yh (T ) being the conforming P1 finite element space. Later, Rodríguez, Hiptmair, and Valli [49] generalized this to k = 0 and d = 3, where Yh (T ) is then the Nédélec finite element space. Discrete Helmholtz decompositions arose also in the context of the Stokes equations (resp. linear elasticity and the biharmonic equation), where deviatoric, i.e. trace-free, (resp. symmetric) tensor fields are decomposed. The first contribution of this paper is an overview of all known discrete Helmholtz decompositions. Since mixed boundary conditions are not much treated in the literature, this paper exemplifies the generalization to mixed boundary conditions for the decompositions of [2, 49]. In 2D, the gradient and the (vector-valued) rot (or Curl) operator are the same up to a change of coordinates and therefore the spaces X h (T ) and Yh (T ) can be interchanged. However, this is not the case in 3D and therefore the decomposition (1.2) with a conforming space X h (T ) is new; cf. Theorem 4.1 below. The third and main contribution of this paper consists of completely new discrete Helmholtz decompositions of piecewise affine vector fields in Theorems 5.1, 5.5 and 5.7. While the decompositions for k = 0 are conforming in one of the spaces X h (T ) and Yh (T ), the decompositions for k = 1 require nonconforming spaces for both X h (T ) and Yh (T ). In 2D these spaces are the Fortin–Soulie spaces, while in 3D the rotation space consists of a Nédélec space enriched by nonconforming bubbles. While in 2D, the decomposition (1.2) follows by the orthogonality of the spaces and a dimension argument, the decomposition (1.2) for k = 1 and d = 3 requires a thorough analysis of the kernel of the operator rot NC . The majority of proofs in this paper focus on the case of contractible domains. However, the presence of handles (multiple connectedness) and cavities (non-connected boundary) in the domain as well as the type of boundary conditions may require 123 Foundations of Computational Mathematics the inclusion of the additional finite-dimensional space of Dirichlet or Neumann fields into the Helmholtz decomposition (1.1). Several remarks in this work address the corresponding generalizations of the discrete decompositions to basic non-contractible domains. The presentation of results for arbitrary geometries and mixed boundary conditions to its full extent is beyond the scope of this paper. Sections 6 and 7 show how the discrete Helmholtz decompositions (1.1) can be generalized to decompositions of tensor fields of deviatoric and symmetric matrices. Those tensor fields arise in the context of the Stokes equations in the case of deviatoric matrices and in the context of linear elasticity and the biharmonic equation for symmetric matrices. For a comprehensive overview of all discrete Helmholtz decompositions of this paper, see Table 1. This table refers to the respective theorems of this paper and also to the literature for previously established results. Discrete Helmholtz decompositions are applied in many different contexts. The discrete Helmholtz decomposition provides the basis for the derivation of stable discretizations for a variety of problems. The first discrete Helmholtz decomposition arose in the analysis of a nonconforming discretization of the Reissner–Mindlin plate [2]. While the decomposition (1.1) allows to treat the continuous problem, a discrete counterpart in [2] mimics the continuous analysis and enables a robust discretization of the problem. This approach was generalized in [32, 33] to arbitrary polynomial degrees. The latter works are based on a discrete Helmholtz decomposition of the form Pk (T ; Rd ) = Z h  ⊥ rot NC Yh (T ) without specifying the space Z h as a space of piecewise gradients. See also the works [50–52] for discretizations based on this kind of discre (...truncated)