Seminar in the framework of the PDE Afternoon



14.11.2017
Type seminar
Time 15:00 - 15:45
Place University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz, 1, Vienna (map)
Room From 15:00 to 15:45: Hörsaal 2 (ground floor) - from 16:00 to 17:00: WPI Seminarraum (8th floor)
Speaker Silvia Bertoluzza (CNR-IMATI Pavia, Italy)
Title Fictitious domain and high order discretization: the Fat Boundary Method
Abstract The Fat Boundary Method (FBM), introduced by Maury in 2001, is a fictitious domain method for solving partial differential equations in a domain with holes. The typical situation, which is met for instance in the context of fluid particle flows, is that of a perforated domain, $\Omega = D \setminus B$, where $D$ is a simple shaped domain, say a cube, and $B$ is a collection of (possibly many) smooth open subsets (the holes). The method consists in splitting the initial equations into two problems to be coupled via Schwartz type iterations: the solution of a global problem set in $D$, for which we assume that fast solvers can be used, and the solution, fully in parallel, of a collection of independent local problems defined on an auxiliary domain $\omega$ composed by narrow strips around the connected components of $B$ (the so called fat boundary). The coupling between the global problem and the local ones is based on the one hand on the interpolation of a globally defined field on the artificial boundary, which together with $\partial B$ delimits the auxiliary domain $\omega$, and on the other hand on the prescription of a jump in the normal derivative across the boundary of $B$. While most fictitious domain methods result in a degradation in the accuracy in comparison with boundary fitted methods, under suitable assumptions the FBM retains optimality, even when using high order discretizations. More precisely it is possible to show that
if the solution $\textrm{u}$ is sufficiently smooth in $\Omega$, domain of definition of the original problem
, then the FBM achieves the best order of approximation allowed by the chosen approximation spaces.

For the class of applications that we have in mind (such as the simulation of blood flow, where the holes are given by the red blood cells) it is however fundamental to consider a wider class of equations (in particular the Navier-Stokes equation) and to take into account, in the analysis, the effect of the approximate computation of the integrals involved in the coupling between the local and the global problem. In this talk we will present the extension of the above method to the Stokes problem (a crucial step for the solution of the Navier-Stokes equation) as well as some preliminary results related with the effect of numerical quadrature on stability and convergence.