Generating admissible space-time meshes for moving domains in d+1d+1-dimensions

In this paper we present a discontinuous Galerkin finite element method for the solution of the transient Stokes equations on moving domains. For the discretization we use an interior penalty Galerkin approach in space, and an upwind technique in time. The method is based on a decomposition of the space-time cylinder into finite elements. Our focus lies on three-dimensional moving geometries, thus we need to triangulate four dimensional objects. For this we will present an algorithm to generate $d+1$-dimensional simplex space-time meshes and we show under natural assumptions that the resulting space-time meshes are admissible. Further we will show how one can generate a four-dimensional object resolving the domain movement. First numerical results for the transient Stokes equations on triangulations generated with the newly developed meshing algorithm are presented

The auxiliary space preconditioner for the de Rham complex

We generalize the construction and analysis of auxiliary space preconditioners to the n-dimensional finite element subcomplex of the de Rham complex. These preconditioners are based on a generalization of a decomposition of Sobolev space functions into a regular part and a potential. A discrete version is easily established using the tools of finite element exterior calculus. We then discuss the four-dimensional de Rham complex in detail. By identifying forms in four dimensions (4D) with simple proxies, form operations are written out in terms of familiar algebraic operations on matrices, vectors, and scalars. This provides the basis for our implementation of the preconditioners in 4D. Extensive numerical experiments illustrate their performance, practical scalability, and parameter robustness, all in accordance with the theory

Space-Time Isogeometric Analysis of Parabolic Evolution Equations

We present and analyze a new stable space-time Isogeometric Analysis (IgA) method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains. The discrete bilinear form is elliptic on the IgA space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields an a priori discretization error estimate with respect to the discrete norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IgA spaces

Space-Time Discretizations Using Constrained First-Order System Least Squares (CFOSLS)

This paper studies finite element discretizations for three types of time-dependent PDEs, namely heat equation, scalar conservation law and wave equation, which we reformulate as first order systems in a least-squares setting subject to a space-time conservation constraint (coming from the original PDE). Available piece- wise polynomial finite element spaces in (n + 1)-dimensions for functional spaces from the (n + 1)-dimensional de Rham sequence for n = 3, 4 are used for the implementation of the method. Computational results illustrating the error behavior, iteration counts and performance of block-diagonal and monolithic geometric multi- grid preconditioners are presented for the discrete CFOSLS system. The results are obtained from a parallel implementation of the methods for which we report reasonable scalability

Molecular regulation of alternative polyadenylation (APA) within the Drosophila nervous system

Alternative polyadenylation (APA) is a widespread gene regulatory mechanism that generates mRNAs with different 3′-ends, allowing them to interact with different sets of RNA regulators such as microRNAs and RNA-binding proteins. Recent studies have shown that during development, neural tissues produce mRNAs with particularly long 3′UTRs, suggesting that such extensions might be important for neural development and function. Despite this, the mechanisms underlying neural APA are not well understood. Here, we investigate this problem within the Drosophila nervous system, focusing on the roles played by general cleavage and polyadenylation factors (CPA factors). In particular, we examine the model that modulations in CPA factor concentration may affect APA during development. For this, we first analyse the expression of the Drosophila orthologues of all mammalian CPA factors and note that their expression decreases during embryogenesis. In contrast to this global developmental decrease in CPA factor expression, we see that cleavage factor I (CFI) expression is actually elevated in the late embryonic central nervous system, suggesting that CFI might play a special role in neural tissues. To test this, we use the UAS/Gal4 system to deplete CFI proteins from neural tissue and observe that in this condition, multiple genes switch their APA patterns, demonstrating a role of CFI in APA control during Drosophila neural development. Furthermore, analysis of genes with 3′UTR extensions of different length leads us to suggest a novel relation between 3′UTR length and sensitivity to CPA factor expression. Our work thus contributes to the understanding of the mechanisms of APA control within the developing central nervous system

Telophase Correction Refines Division Orientation in Stratified Epithelia

During organogenesis, precise control of spindle orientation balances proliferation and differentiation. In the developing murine epidermis, planar and perpendicular divisions yield symmetric and asymmetric fate outcomes, respectively. Classically, division axis specification involves centrosome migration and spindle rotation, events occurring early in mitosis. Here, we identify a novel orientation mechanism which corrects erroneous anaphase orientations during telophase. The directionality of reorientation correlates with the maintenance or loss of basal contact by the apical daughter. While the scaffolding protein LGN is known to determine initial spindle positioning, we show that LGN also functions during telophase to reorient oblique divisions toward perpendicular. The fidelity of telophase correction also relies on the tension-sensitive adherens junction proteins vinculin, α-E-catenin, and afadin. Failure of this corrective mechanism impacts tissue architecture, as persistent oblique divisions induce precocious, sustained differentiation. The division orientation plasticity provided by telophase correction may enable progenitors to adapt to local tissue needs