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Finite elements for symmetric tensors in three dimension
Douglas N. Arnold, Gerard Awanou, Ragnar Winther
TL;DR
Three-dimensional Hellinger–Reissner elasticity lacked known stable mixed finite elements using discontinuous polynomial displacements. The paper constructs tetrahedral symmetric-tensor H(div) spaces for every displacement degree and proves their stability through commuting projections and elasticity-complex structure. The resulting family includes a lowest-order stress space of local dimension 162, while element complexity may limit practical significance.
Problem
Stable mixed finite elements for the three-dimensional Hellinger–Reissner formulation were not known, making an appropriate symmetric-tensor H(div) stress discretization a longstanding challenge.
Method
The paper constructs tetrahedral finite element spaces Σh and Vh for each polynomial degree and analyzes them through the elasticity complex and commuting projections.
Results
The constructed spaces satisfy the two stated mixed-method stability conditions for every polynomial degree k ≥1; for k = 1, the local stress space has dimension 162.
Takeaways & Limitations
The construction provides stable three-dimensional mixed discretizations and insight into obstacles to simpler or nonconforming methods.
Takeaways & Limitations
The complexity of the elements may limit their practical significance.
Abstract
from arXiv · showhide
We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger--Reissner mixed formulation of the elasticty equations, when standard discontinous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex, and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.
DIMENSIONS
The passage identifies the paper’s authors.
- The authors are Douglas Arnold, Gerard Awanou, and Ragnar Winther.
1. Introduction
The paper develops stable three-dimensional mixed finite elements for elasticity by constructing symmetric-tensor H(div) spaces on tetrahedral meshes. Its elements satisfy mixed-method stability conditions, although their complexity may limit practical significance.
- Motivation: The paper addresses the challenge of constructing stable H(div, Ω; S) stress spaces for three-dimensional Hellinger–Reissner elasticity.Displacement is approximated by discontinuous piecewise polynomials, while stress requires symmetric tensors with square-integrable divergence.
- Elasticity complex: The construction forms a finite subcomplex of the elasticity complex and is related to it by commuting diagrams.The analysis also connects symmetric H(div) discretization with H(curl curl∗, Ω; S) through a second-order operator.
- Construction: The authors construct tetrahedral finite element families Σh and Vh, with one family member for each polynomial degree k ≥1.Vh consists of piecewise polynomial vector fields of degree at most k.
- Stability: The spaces satisfy div Σh ⊂ Vh and admit a uniformly bounded projection commuting with divergence.These are sufficient stability conditions for the discrete mixed system.
- Lowest order element: For k = 1, the local stress space has dimension 162 and augments quadratic polynomials with divergence-free polynomials of degrees 3 and 4.This three-dimensional lowest-order space has 27 degrees of freedom per stress component on average.
- Limitations: The complexity of the elements may limit their practical significance, despite the insight they provide for simpler or nonconforming methods.
2. Notation and preliminaries
The preliminaries establish notation, polynomial complexes, mesh assumptions, and the analytical framework supporting the finite element construction. Exactness of the elasticity-related complexes and commuting projections underpin stability.
- Simplex notation: For a tetrahedron K, Δ2(K), Δ1(K), and Δ0(K) denote its faces, edges, and vertices, respectively.The notation Δ(K) collects all subsimplices of dimensions 0 through 3.
- Elasticity complex: The elasticity complex uses symmetric and skew-symmetric matrix spaces, infinitesimal rigid motions, and differential operators including curl curl∗.On contractible domains, the elasticity complex is exact.
- Polynomial complex: The polynomial elasticity complex is also exact, with polynomial kernels and images linked through the symmetric gradient, divergence, and curl curl∗.The proof constructs polynomial potentials for curl curl∗-free and divergence-free symmetric fields.
- Mesh assumptions: The mesh consists of shape-regular tetrahedral triangulations whose tetrahedra meet in common subsimplices and have diameters bounded by h.
- Discrete framework: The discrete spaces satisfy div Σh ⊂ Vh and use a projection Πh commuting with divergence; these properties imply stability and error bounds.The projection is uniformly bounded with respect to mesh size.
3. The lowest order element
The lowest-order tetrahedral pair uses quadratic symmetric stresses augmented by divergence-free cubic and quartic fields, with displacement space P1 and stable mixed discretization properties.
- The local displacement space is VK = P1(K; R3), while ΣK contains fourth-degree symmetric tensors whose divergence lies in P1(K; R3).
- The lowest-order stress space has dimension 162, obtained from P4(K; S) subject to 48 divergence constraints.
- The space M(K) consists of quartic symmetric tensors with zero divergence and zero normal trace on the tetrahedron boundary, and has dimension 6.
- A unisolvent set of degrees of freedom combines vertex values, edge moments, face moments, element averages, and moments against M(K).
- Global continuity of normal components across faces places Σh in H(div, Ω; S), while additional edge continuity follows from the degrees of freedom.
- The canonical interpolant is not H1-bounded, so a modified interpolant is introduced; the resulting pair is stable and supports the stated error estimates.
4. A Family of Higher Order Elements
The construction extends to one stable tetrahedral element pair for every polynomial degree k ≥ 1, with degree-dependent stress spaces, degrees of freedom, interpolation estimates, and error analysis.
- For every k ≥ 1, the displacement space is piecewise Pk(K; R3), paired with stress fields in Pk+3(K; S) whose divergence lies in Pk(K; R3).
- The local stress-space dimension is bounded below by dk = k^3 + 12k^2 + 56k + 93.
- Higher-order degrees of freedom use vertex values, edge and face moments, and interior moments against strain and divergence-free polynomial spaces.
- The degrees of freedom are unisolvent, their count equals dk, and explicit bases for Mk(K) are supplied for k = 4 and k = 5.
- The modified canonical interpolant satisfies the commuting stability conditions and the approximation bound ∥ΠhT − T∥0 ≤ ch^m∥T∥m for 1 ≤ m ≤ k + 2.
5. Some properties of vector fields and matrix fields
This section develops vector- and matrix-field identities, tangential differential operators, exact two-dimensional complexes, and integration-by-parts tools for analyzing symmetric tensor complexes.
- The projections Pn and Qn split vectors and matrix fields into normal and tangential components relative to a plane with unit normal n.
- Tangential operators gradf, curlf, and rotf are defined intrinsically on the plane, with matrix versions applied row-wise or column-wise as appropriate.
- The decomposition of curl and curl∗ into normal and tangential parts supports the matrix identities used later for symmetric fields.
- Exact two-dimensional complexes are introduced on planes, paralleling the three-dimensional elasticity-complex construction.
- Integration-by-parts formulas relate curl curl∗ and boundary terms involving tangential differential operators and normal traces.
- A piecewise smooth symmetric field belongs to H(curl curl∗, Ω; S) exactly when QnSQn and Λf(S) are continuous across every mesh face.
6. Polynomial matrix fields on a single tetrahedron
The section develops polynomial symmetric-tensor spaces on a tetrahedron using face, edge, and barycentric-coordinate structure, then derives their dimensions and inter-face compatibility properties.
- Setup: The analysis is restricted to one tetrahedron and supports a discrete elasticity complex with piecewise polynomial spaces on a triangulation.The resulting construction is used later to define the global space Θh.
- Constrained spaces: The constrained space Nk consists of degree-k symmetric polynomial fields whose face-normal components QnSQn and Λf(S) vanish on every face.The space Ng is characterized by vanishing QnSQn on all faces containing the subsimplex g.
- Polynomial representation: Barycentric coordinates provide a unique organization of polynomial components by the subsimplexes whose coordinates enter each monomial.This organizes polynomial representations and supports the associated degrees of freedom.
- Dimension and compatibility: Across a shared edge, the face-derived quantities Tf have matching tangential components, yielding an inter-face compatibility condition.Specifically, s′Tf+(U)s = s′Tf−(U)s on the common edge.
- Dimension and compatibility: For k ≥3, dim Nk = k(k^2 −6k + 11).The dimension follows after establishing that elements of Nk vanish on each edge and analyzing the remaining face data and compatibility conditions.
7. The space of divergence-free matrix fields with vanishing normal traces Recall that the space
The section studies divergence-free symmetric polynomial fields with vanishing normal traces, relates them to curl curl∗ images through an exact sequence, and computes their dimension.
- Definition and role: The space Mk contains degree-k symmetric polynomial fields that are divergence-free on K and have vanishing normal traces on every face.It is central to the finite element space Σh, for which its dimension and a basis are needed.
- Exact-sequence structure: The operator curl curl∗ maps Nk+2(K) onto Mk(K), while its kernel is described through symmetric gradients of boundary-vanishing vector polynomials.These relationships form an exact sequence used to analyze Mk.
- Proof strategy: The proof constructs a vector field whose face restrictions reproduce the tangential data of a polynomial tensor, then adjusts it to obtain the required boundary relation.The construction uses facewise potentials and an additional vector field with prescribed normal traces.
- Dimension: For k ≥4, dim Mk(K) = (k + 2)(k −2)(k −3)/2.The formula follows from the short exact sequence and the dimension formula for Nk.
- Basis construction: Explicit bases are constructed for Mk when k = 4 and k = 5, with computational algebra suggested for other degrees.For k = 4, a basis is obtained from a basis of N^0_2; for k = 5, the curl curl∗ kernel has dimension 3 and injectivity is recovered on a codimension-3 subspace.
8. A Discrete Elasticity Complex
The construction embeds the finite element spaces in a discrete elasticity complex. The section defines the auxiliary spaces, their degrees of freedom, continuity properties, and interpolation operators.
- The spaces Vh and Σh are constituents of a discrete elasticity complex, revealing additional structure in their construction.
- The complex can use any polynomial-level pair for Σh and Vh, although the discussion specializes to piecewise quartic stresses with linear divergence and piecewise linear displacements.
- The auxiliary space Θh is locally P6(K; S), has dimension 504, and is specified by 504 degrees of freedom.
- The degrees of freedom are shown to be unisolvent for P6(K; S), while the associated interpolation identities verify that the discrete complex diagram commutes.
- The degrees of freedom make Θh a subspace of H(curl curl∗, Ω; S) and define a canonical interpolation operator reproducing those functionals.
- The auxiliary space Wh is locally P7(K; R3), has dimension 360, and its continuity conditions yield face continuity and C1 regularity at vertices.