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The Elasticity Complex: Compact Embeddings and Regular Decompositions
TL;DR: A simple technique is presented to prove the compact embeddings based on regular decompositions/potentials and Rellich's section theorem, which can be easily adapted to any Hilbert complex.
Abstract: We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincare type estimates, Helmholtz type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's section theorem, which can be easily adapted to any Hilbert complex.
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TL;DR: In this article, the authors revisited the construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and applied it to particular choices of differential operators, and the resulting index is then computed with the help of explicitly describing the dimension of the cohomology groups of generalised Dirichlet and Neumann tensor fields.
Abstract: We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the help of explicitly describing the dimension of the cohomology groups of generalised (`harmonic') Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac-type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and for the theory of general relativity. The differential operators are of mixed order and cannot be seen as leading order type with relatively compact perturbation. As a by-product we present a comprehensive description of the underlying generalised Dirichlet-Neumann vector and tensor fields defining the respective cohomology groups, including an explicit construction of bases in terms of topological invariants, which are of both analytical and numerical interest. Though being defined by certain projection mechanisms, we shall present a way of computing these basis functions by solving certain PDEs given in variational form. For all of this we rephrase core arguments in the work of Rainer Picard [1982] applied to the de Rham complex and use them as a blueprint for the more involved cases presented here. In passing, we also provide new vector-analytical estimates of generalised Poincare-Friedrichs type useful for elasticity or the theory of general relativity.
10 citations
TL;DR: In this paper , the Hilbert complex of elasticity involving spaces of symmetric tensor fields was investigated and closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decomposition, regular potentials, finite cohomology groups, and compact embedding results were obtained.
Abstract: We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's selection theorem, which can be easily adapted to any Hilbert complex.
7 citations
24 Jul 2022
TL;DR: In this paper , it was shown that the biharmonic Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact, and the results of Pauly and Zulehner on the de Rham and elasticity Hilbert complexes were also proved.
Abstract: We show that the biharmonic Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. This paper extends recent results of the authors on the de Rham and elasticity Hilbert complexes with mixed boundary conditions and results of Pauly and Zulehner on the biharmonic Hilbert complex with empty or full boundary conditions.
3 citations
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TL;DR: In this paper, the de Rham Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is shown to be closed and compact, and the crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions.
Abstract: We show that the de Rham Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are proved as well.
2 citations
TL;DR: In this paper , the authors revisited the construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and applied it to particular choices of differential operators, and computed the indices of Dirac type operators associated to the elasticity complex and the newly found biharmonic complex.
Abstract: Abstract We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed using an explicit description of the cohomology groups of generalised (‘harmonic’) Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and for the theory of general relativity. The differential operators are of mixed order and cannot be seen as leading order type with relatively compact perturbation. As a by-product we present a comprehensive description of the underlying generalised Dirichlet–Neumann vector and tensor fields defining the respective cohomology groups, including an explicit construction of bases in terms of topological invariants, which are of both analytical and numerical interest. Though being defined by certain projection mechanisms, we shall present a way of computing these basis functions by solving certain PDEs given in variational form. For all of this we rephrase core arguments in the work of Rainer Picard [42] applied to the de Rham complex and use them as a blueprint for the more involved cases presented here. In passing, we also provide new vector-analytical estimates of generalised Poincaré–Friedrichs type useful for elasticity or the theory of general relativity.
2 citations
References
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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
"The Elasticity Complex: Compact Emb..." refers background in this paper
...Among others, we cite the important works in [3], [4], [6, 5, 7, 8], [10], [16, 17] [1, 2], [14], [18, 19] as well as the vast literature cited therein....
[...]
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: In this article, the authors developed an abstract Hilbert space framework for analyzing stability and convergence of finite element approximations of the Hodge Laplacian in 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 the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, 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 to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.
436 citations
"The Elasticity Complex: Compact Emb..." refers background in this paper
...Among others, we cite the important works in [3], [4], [6, 5, 7, 8], [10], [16, 17] [1, 2], [14], [18, 19] as well as the vast literature cited therein....
[...]
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
"The Elasticity Complex: Compact Emb..." refers background in this paper
...Among others, we cite the important works in [3], [4], [6, 5, 7, 8], [10], [16, 17] [1, 2], [14], [18, 19] as well as the vast literature cited therein....
[...]
TL;DR: In this article, the authors give a proof for a large class of bounded domains of the following compactness statements: if G is a bounded domain, β is a tensor-valued function on G satisfying certain restrictions, and if β is interpreted as electric dielectricity ϵ or as magnetic permeability μ, then β has a L2-convergent subsequence subsequence.
Abstract: The paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: Let G be a bounded domain, β be a tensor-valued function on G satisfying certain restrictions, and let {n} be a sequence of vector-valued functions on G where the L2-norms of {n}, {curl n}, and {div(β n)} are bounded, and where all n either satisfy x n = 0 or (β Fn) = 0 at the boundary ∂G of G ( = normal to ∂G): then {n} has a L2-convergent subsequence The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ∂G if β is interpreted as electric dielectricity ϵ or as magnetic permeability μ, respectively
These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for Maxwell's equations
275 citations
"The Elasticity Complex: Compact Emb..." refers background or result in this paper
...Here we cite results from [30, 15], and [25], see also the corresponding literature in [25]....
[...]
...In Weck’s work [31] one finds the first result for piece-wise smooth domains, and in Weber’s paper [30] the first proof being valid for strong Lipschitz domains is given....
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