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The Elasticity Complex: Compact Embeddings and Regular Decompositions
Dirk Pauly,Walter Zulehner +1 more
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TLDR
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.read more
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A Reissner-Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations
TL;DR: In this article , a finite element discretisation of the shear-deformable Reissner-Mindlin plate problem based on the Hellinger-Reissner principle of symmetric stresses is presented.
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Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint
TL;DR: In this article , a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces, and the theory of boundary control system is employed to deal with problems from mechanics that cannot be handled by the geometric setting.
Families of Annihilating Skew-Selfadjoint Operators and their Connection to Hilbert Complexes
Dirk Pauly,Rainer Picard +1 more
TL;DR: In this paper , it was shown that Hilbert complexes are strongly related to annihilating sets of skew-selfadjoint operators, which is a new perspective on the classical topic of Hilbert complexes viewed as families of normal operators.
References
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Journal ArticleDOI
Finite element exterior calculus, homological techniques, and applications
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.
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Finite element exterior calculus: From hodge theory to numerical stability
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.
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Finite element exterior calculus: from Hodge theory to numerical stability
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.
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Mixed finite elements for elasticity
Douglas N. Arnold,Ragnar Winther +1 more
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.
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A local compactness theorem for Maxwell's equations
Ch. Weber,P. Werner +1 more
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.
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