Local bounded cochain projections FALK, RICHARD S.; WINTHER, RAGNAR
Mathematics of computation,
11/2014, Letnik:
83, Številka:
290
Journal Article
Recenzirano
Odprti dostop
We construct projections from H \Lambda ^k(\Omega ) forms on \Omega L^2(\Omega ) L^2(\Omega ) H \Lambda ^k(\Omega ). Thus, their definition requires less smoothness than assumed for the definition of ...the canonical interpolants based on the degrees of freedom. Moreover, these projections have the properties that they commute with the exterior derivative and are bounded in the H \Lambda ^k(\Omega ). Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Clément interpolant. A double complex structure is introduced as a key tool to carry out the construction.
A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an
n
n
simplex to the interior is to use so-called rational blending functions. The ...purpose of this paper is to generalize the construction by blending to the de Rham complex. More precisely, we define polynomial preserving extensions which map traces of
k
k
-forms defined on the boundary of the simplex to
k
k
-forms defined in the interior. Furthermore, the extensions are cochain maps, i.e., they commute with the exterior derivative.
Two families of conforming finite elements for the two-dimensional Stokes problem are developed, guided by two discrete smoothed de Rham complexes, which we coin "Stokes complexes." We show that the ...finite element pairs are inf-sup stable and also provide pointwise mass conservation on very general triangular meshes.
Basic principles of mixed Virtual Element Methods Brezzi, F.; Falk, Richard S.; Donatella Marini, L.
ESAIM. Mathematical modelling and numerical analysis,
07/2014, Letnik:
48, Številka:
4
Journal Article
Recenzirano
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The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n ...− 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).
A coding variant of the inflammatory bowel disease (IBD) risk gene
has been associated with defective autophagy and deregulation of endoplasmic reticulum (ER) function. IL-22 is a barrier protective ...cytokine by inducing regeneration and antimicrobial responses in the intestinal mucosa. We show that ATG16L1 critically orchestrates IL-22 signaling in the intestinal epithelium. IL-22 stimulation physiologically leads to transient ER stress and subsequent activation of STING-dependent type I interferon (IFN-I) signaling, which is augmented in
intestinal organoids. IFN-I signals amplify epithelial TNF production downstream of IL-22 and contribute to necroptotic cell death. In vivo
IL-22 treatment in
and
/
mice potentiates endogenous ileal inflammation and causes widespread necroptotic epithelial cell death. Therapeutic blockade of IFN-I signaling ameliorates IL-22-induced ileal inflammation in
mice. Our data demonstrate an unexpected role of
in coordinating the outcome of IL-22 signaling in the intestinal epithelium.
The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) ...for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh
T
of the domain
Ω
, we build a map which decomposes piecewise smooth
k
-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of
k
-forms. Furthermore, the transform is bounded in
L
2
and also on the appropriate subspace consisting of
k
-forms with exterior derivatives in
L
2
.
In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and ...displacements. The methods are based on a modified form of the Hellinger--Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.
A plethora of functional and genetic studies have suggested a key role for the IL-23 pathway in chronic intestinal inflammation. Currently, pathogenic actions of IL-23 have been ascribed to specific ...effects on immune cells. Herein, we unveil a protective role of IL-23R signaling. Mice deficient in IL-23R expression in intestinal epithelial cells (Il23RΔIEC) have reduced Reg3b expression, show a disturbed colonic microflora with an expansion of flagellated bacteria, and succumb to DSS colitis. Surprisingly, Il23RΔIEC mice show impaired mucosal IL-22 induction in response to IL-23. αThy-1 treatment significantly deteriorates colitis in Il23RΔIEC animals, which can be rescued by IL-22 application. Importantly, exogenous Reg3b administration rescues DSS-treated Il23RΔIEC mice by recruiting neutrophils as IL-22-producing cells, thereby restoring mucosal IL-22 levels. The study identifies a critical barrier-protective immune pathway that originates from, and is orchestrated by, IL-23R signaling in intestinal epithelial cells.
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•IL-23R transduces signals into the intestinal epithelium•Il23RΔIEC mice are susceptible to DSS colitis and have a disturbed gut microflora•Epithelial IL-23R is required for optimal secretion of the c-type lectin Reg3b•The c-type lectin Reg3b promotes recruitment of IL-22-producing cells as an alarmin•Systemic substitution of Reg3b rescues the gut barrier defect of Il23RΔIEC mice
Aden et al. show that epithelial IL-23R signaling initiates a Reg3b-dependent chemoattraction of IL-22-producing neutrophil granulocytes into the intestinal lamina propria, limiting flagellated bacteria content and intestinal inflammation.
We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical ...finite element spaces for vector fields, frequently referred to as Raviart–Thomas, Brezzi–Douglas–Marini, and Nédélec spaces. In the present paper, we derive geometric decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis.
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