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A finite element elasticity complex in three dimensions.
TL;DR: In this article, a finite element elasticity complex on tetrahedral meshes is devised and the key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the $\textrm{inc}$ operator.
Abstract: A finite element elasticity complex on tetrahedral meshes is devised. The $H^1$ conforming finite element is the smooth finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an $H(\textrm{inc})$-conforming finite element for symmetric tensors is the main focus of this paper. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the $\textrm{inc}$ operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition of polynomial tensor spaces. The trace of the $\textrm{inc}$ operator is induced from a Green's identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Our construction appears to be the first $H(\textrm{inc})$-conforming finite elements on tetrahedral meshes without further splits.
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TL;DR: Two types of finite element spaces on a tetrahedron are constructed for divdiv conforming symmetric tensors in three dimensions and several decomposition of polynomial vector and tensors spaces are revealed from the complexes.
Abstract: Two types of finite element spaces on a tetrahedron are constructed for divdiv conforming symmetric tensors in three dimensions. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace operators. First, the divdiv Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensors spaces are derived from the polynomial complexes. Then, traces for div-div operator are characterized through a Green's identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Third, a finite element for sym curl-conforming trace-free tensors is constructed following the same approach. Finally, a finite element divdiv complex, as well as the bubble functions complex, in three dimensions are established.
21 citations
01 Jan 2022
TL;DR: In this paper , a family of conforming virtual element Hessian complexes on tetrahedral meshes is constructed based on decompositions of polynomial tensor spaces and applied to discretize the linearized time-independent Einstein-Bianchi system with optimal order convergence.
Abstract: A family of conforming virtual element Hessian complexes on tetrahedral meshes are constructed based on decompositions of polynomial tensor spaces. They are applied to discretize the linearized time-independent Einstein-Bianchi system with optimal order convergence.
4 citations
TL;DR: In this article , two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the non-conforming P1-P0 element for the Stokes equation in three dimensions are studied.
Abstract: Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming P1-P0 element for the Stokes equation in three dimensions are studied. Commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order H(gradcurl)-nonconforming finite element only has 14 degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The H(gradcurl)-nonconforming elements are applied to solve the quad-curl problem, and the optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming P1-P0 element method for the Stokes equation, based on which a fast solver is discussed. Numerical results are provided to verify the theoretical convergence rates.
3 citations
TL;DR: In this paper , the shape function space is first split as the trace space and the bubble space, and the later is further decomposed into the null space of the differential operator and its orthogonal complement.
Abstract: Several div-conforming and divdiv-conforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed in this work. The shape function space is first split as the trace space and the bubble space. The later is further decomposed into the null space of the differential operator and its orthogonal complement. Instead of characterizations of these subspaces of the shape function space, characterizations of corresponding degrees of freedom in the dual spaces are provided. Vector div-conforming finite elements are first constructed as an introductory example. Then new symmetric div-conforming finite elements are constructed. The dual subspaces are then used as build blocks to construct new divdiv-conforming finite elements.
2 citations
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TL;DR: Several div-conforming and divdivconforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed in this article, where the shape function space is first split as the trace space and the bubble space.
Abstract: Several div-conforming and divdiv-conforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed in this work. The shape function space is first split as the trace space and the bubble space. The later is further decomposed into the null space of the differential operator and its orthogonal complement. Instead of characterization of these subspaces of the shape function space, characterization of the duals spaces are provided. Vector div-conforming finite elements are firstly constructed as an introductory example. Then new symmetric div-conforming finite elements are constructed. The dual subspaces are then used as build blocks to construct divdiv conforming finite elements.
1 citations
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TL;DR: Finite element exterior calculus as mentioned in this paper is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations, which brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological and algebraic structures which underlie well-posedness of the PDE problem being solved.
Abstract: Finite element exterior calculus is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are re- vealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Lapla- cian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.
1,044 citations
TL;DR: In this article, the authors consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem.
Abstract: This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodgetheoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.
550 citations
TL;DR: The elements presented here are the first ones using polynomial shape functions which are known to be stable, and show stability and optimal order approximation.
Abstract: There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available.
393 citations
TL;DR: Finite element subspaces of the space of symme- tric tensors with square-integrable divergence on a three-dimensional domain are constructed, which can be viewed as the three- dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.
Abstract: We construct finite element subspaces of the space of symme- tric 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 stan- dard discontinous finite element spaces are used to approximate the displace- ment 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.
171 citations