Hilbert complexes with mixed boundary conditions part 1: de Rham complex

Pauly, Dirk; Schomburg, Michael

doi:10.1002/mma.7894

Cited by 15 publications

(51 citation statements)

References 18 publications

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“…Higher Sobolev order results are also proved. This paper extends recent results on the de Rham and elasticity Hilbert complexes with mixed boundary conditions from [3,4] and results on the biharmonic Hilbert complex with empty or full boundary conditions from [6].…”

supporting

confidence: 83%

“…We shall follow in close lines the rationale from [3,4]. The main results comprise regular decompositions and regular potentials, compact embeddings, Helmholtz decompositions, closed ranges, Friedrichs/Poincaré type estimates, and bases of the corresponding cohomology groups (generalised Dirichlet/Neumann tensors).…”

Section: Introductionmentioning

confidence: 91%

“…In [3] we investigated the de Rham Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains…”

Section: Introductionmentioning

confidence: 99%

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Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes

Pauly¹,

Schomburg²

2022

Preprint

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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 on the de Rham and elasticity Hilbert complexes with mixed boundary conditions from [3,4] and results on the biharmonic Hilbert complex with empty or full boundary conditions from [6].

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confidence: 83%

Section: Introductionmentioning

confidence: 91%

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Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes

Pauly¹,

Schomburg²

2022

Preprint

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“…For the sake of clarity we write H( Ω, div, S) and H Γ1 ( Ω, div, S) for the corresponding spaces on the parametric domain. For more information about such spaces with boundary conditions we refer to the articles [20,21]. In latter references the explanations are for the three-dimensional case, but they can be adapted easily to planar domains.…”

Section: An Auxiliary Mixed Bc Problemmentioning

confidence: 99%

Mixed Isogeometric Discretizations for Planar Linear Elasticity

Jeremias¹,

Simeon²

2022

Preprint

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In this article we suggest two discretization methods based on isogeometric analysis (IGA) for planar linear elasticity. On the one hand, we apply the well-known ansatz of weakly imposed symmetry for the stress tensor and obtain a well-posed mixed formulation. Such modified mixed problems have been already studied by different authors. But we concentrate on the exploitation of IGA results to handle also curved boundary geometries. On the other hand, we consider the more complicated situation of strong symmetry, i.e. we discretize the mixed weak form determined by the so-called Hellinger-Reissner variational principle. We show the existence of suitable approximate fields leading to an inf-sup stable saddle-point problem. For both discretization approaches we prove convergence statements and in case of weak symmetry we illustrate the approximation behavior by means of several numerical experiments.

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“…The key trace operators and trace spaces associated with the Euclidean de Rham complex in three space dimensions have already been introduced 5 We denote by σ the surface measure on the boundary. 6 For the functional analytic foundations, we refer to parts of the FA-ToolBox from [35,Sec. 2], which is a compilation of useful functional analysis results that grew from its use in previous works, cf.…”

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confidence: 99%

Traces for Hilbert Complexes

Hiptmair¹,

Pauly²,

Schulz³

2022

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We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality relationships between trace spaces. We elaborate that many properties of the classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be formulated using quotient spaces/annihilators. We show that, if a Hilbert complex admits stable "regular decompositions" with compact lifting operators, then the associated trace Hilbert complex is Fredholm. Incarnations of abstract concepts and results in the concrete case of the de Rham complex in three-dimensional Euclidean space will be discussed throughout.

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Hilbert complexes with mixed boundary conditions part 1: de Rham complex

Cited by 15 publications

References 18 publications

Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes

Hilbert Complexes with Mixed Boundary Conditions -- Part 3: Biharmonic Complexes

Mixed Isogeometric Discretizations for Planar Linear Elasticity

Traces for Hilbert Complexes

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