Journal ArticleDOI
Complexes from Complexes
Douglas N. Arnold,Kaibo Hu +1 more
TLDR
A systematic procedure is presented which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old, and shows that the new complex has closed ranges and satisfies a Hodge decomposition, Poincare type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition and compactness properties on general Lipschitz domains.Abstract:
This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincare-type inequalities, well-posed Hodge–Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domainsread more
Citations
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Posted Content
A family of finite element Stokes complexes in three dimensions.
Kaibo Hu,Qian Zhang,Zhimin Zhang +2 more
TL;DR: It is shown that the new elements lead to convergent algorithms for solving a gradcurl model problem as well as solving the Stokes system with precise divergence-free condition and the validity of the algorithms is demonstrated by numerical experiments.
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Finite elements for divdiv-conforming symmetric tensors in three dimensions.
Long Chen,Xuehai Huang +1 more
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.
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The Elasticity Complex: Compact Embeddings and Regular Decompositions
Dirk Pauly,Walter Zulehner +1 more
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.
Journal ArticleDOI
Primal and mixed finite element formulations for the relaxed micromorphic model
TL;DR: In this paper , a relaxed micromorphic model is proposed to model the entire body, including every interior cell, by enriching the kinematics of the mathematical model.
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A discrete elasticity complex on three-dimensional Alfeld splits.
TL;DR: This work constructs conforming finite element elasticity complexes on the Alfeld splits of tetrahedra based on an algebraic machinery that derives the elasticity complex from de~Rham complexes, and smoother finite element differential forms.
References
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Book
Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book
TL;DR: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software.
Journal ArticleDOI
Effects of couple-stresses in linear elasticity
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.
Journal ArticleDOI
Finite elements in computational electromagnetism
TL;DR: In this paper, finite element Galerkin schemes for a number of linear model problems in electromagnetism were discussed, and the finite element schemes were introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms.
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