Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems

Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

A meta-analysis of prenatal diagnosis of congenital malformations of the airways

Congenital malformations of the respiratory tract is an important issue. They cause more than 100 deaths annually in the Russian Federation. The definition and classification of congenital malformations of the respiratory tract are described and developed by Je. A. Cvetkov. The ultrasound is the primal method used in prenatal diagnosis of congenital malformations of the upper respiratory tract. Starting from the age of 17-23 weeks old, the ultrasound shows the following signs: hyperechoic lungs, dilatation of the lower respiratory tract, flattening of the diaphragm. This ultrasound pattern is typical for Congenital High Airway Obstruction Syndrome (CHAOS). If the doctor suspects an identifying of these signs, a follow-up examination is recommended. If the congenital obstruction of the upper respiratory tract is diagnosed during the pregnancy, the childbirth for patients must be performed in the hospital, where it is possible to perform tracheotomy and artificial ventilation of the lungs to the newborn, to operate him with placental support. This type of surgical treatment by the EXIT method (The ex utero intrapartum treatment procedure) is a new stage in neonatal surgery. A meta-analysis of 15 articles in the foreign literature about the detection of CHAOS from 1993 to 2014 was conducted. The analysis includes studies with more than 5 described cases. The statistics of diagnosis and treatment of congenital malformations of the larynx of patients of the ENT clinic of SPbSPMU for the period from 2003 to 2016 with the diagnosis of congenital malformation of the larynx is presented.

Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes

We propose in this work a unified formulation of mixed and primal discretization methods on polyhedral meshes hinging on globally coupled degrees of freedom that are discontinuous polynomials on the mesh skeleton. To emphasize this feature, these methods are referred to here as discontinuous skeletal. As a starting point, we define two families of discretizations corresponding, respectively, to mixed and primal formulations of discontinuous skeletal methods. Each family is uniquely identified by prescribing three polynomial degrees defining the degrees of freedom, and a stabilization bilinear form which has to satisfy two properties of simple verification: stability and polynomial consistency. Several examples of methods available in the recent literature are shown to belong to either one of those families. We then prove new equivalence results that build a bridge between the two families of methods. Precisely, we show that for any mixed method there exists a corresponding equivalent primal method, and the converse is true provided that the gradients are approximated in suitable spaces. A unified convergence analysis is carried out delivering optimal error estimates in both energy- and L2-norms.

An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators

We develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (elementwise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the L2-norm of the potential for smooth solutions. In the lowest-order case, equivalence with the Hybrid Finite Volume method is shown. The theoretical results are confirmed by numerical experiments up to order 4 on several polygonal meshes.