scispace - formally typeset
Journal ArticleDOI: 10.1007/S10208-021-09498-9

Complexes from Complexes

05 Mar 2021-Foundations of Computational Mathematics (Springer US)-Vol. 21, Iss: 6, pp 1739-1774
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 domains

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Topics: Cohomology (52%), Differential equation (52%), Boundary value problem (52%) ... read more
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10 results found


Open accessPosted Content
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.

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Topics: Symmetric tensor (62%), Tensor (62%), Polynomial (57%) ... read more

6 Citations


Open accessPosted Content
Abstract: One conforming and one non-conforming virtual element Hessian complexes on tetrahedral grids are constructed based on decompositions of polynomial tensor space. They are applied to discretize the linearized time-independent Einstein-Bianchi system.

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Topics: Tensor (58%), Hessian matrix (58%), Polynomial (56%)

5 Citations


Open accessPosted Content
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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Topics: Symmetric tensor (56%), Lipschitz continuity (53%), Tensor field (53%) ... read more

4 Citations


Open accessPosted Content
Abstract: We construct finite element Stokes complexes on tetrahedral meshes in three-dimensional space. In the lowest order case, the finite elements in the complex have 4, 18, 16, and 1 degrees of freedom, respectively. As a consequence, we obtain gradcurl-conforming finite elements and inf-sup stable Stokes pairs on tetrahedra which fit into complexes. We show 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. We demonstrate the validity of the algorithms by numerical experiments.

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4 Citations


Open accessPosted Content
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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Topics: Polynomial (hyperelastic model) (60%), Finite element method (56%), Tensor (55%) ... read more

2 Citations


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57 results found


Open accessBook
01 Jan 1990-
Topics: Operator theory (74%), Fourier integral operator (73%), Constant coefficients (69%) ... read more

8,694 Citations


Open accessBook
24 Feb 2012-
Abstract: 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. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

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1,982 Citations



Open accessJournal ArticleDOI: 10.1017/S0962492906210018
01 May 2006-Acta Numerica
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.

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885 Citations


Open accessJournal ArticleDOI: 10.1017/S0962492902000041
Ralf Hiptmair1Institutions (1)
01 Jan 2002-Acta Numerica
Abstract: This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

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Topics: Finite element exterior calculus (68%), Finite element method (61%), Differential form (53%) ... read more

796 Citations