Recent advances in finite element approximations

The Faculty of Informatics is pleased to announce a seminar given by Francesca Rapetti

DATE: Monday, April 4th 2011
PLACE: USI Università della Svizzera italiana, room A22, Red building (Via G. Buffi 13)
TIME: 09.30

ABSTRACT:
Spectral and hp-finite elements are among the most successful high-order methods for the numerical approximation of partial differential equations (PDEs). These methods can achieve spectral convergence of the discrete solution by allowing both h-refinement of the elements mesh and p-refinement of the polynomial degree in each element. The need to extend spectral element formulations to complex geometries and unstructured meshes has recently led to the construction and study of triangular/tetrahedral spectral elements (TSEM); see, e.g., Chen & Babuska, 1995, Hestaven et al., 1998, 2000, 2002, Taylor et al., 2000, etc.. We thus present the TSEM based on the Fekete nodes and some simple applications to Navier-Stokes equations (Pasquetti & Rapetti, since 2004).
To enhance the efficiency of the simulation to treat more realistic applications in sciences and engineering, special numerical tools from domain decomposition are combined with the previous methods. They can refer to optimal nonconforming discretization techniques, to couple different variational approaches, as the mortar element approach (Bernardi, Maday, Patera, 1993), or to efficient iterative solvers (Smith, Bjorstad, Gropp, 1996) and parallelization techniques. We present some applications in elastic wave propagation and coupled magnetomechanical problems with moving conductors. In electromagnetic applications, nodal elements can yield to spurious modes when applied to discretize vector fields, thus a special class of finite elements is generally adopted, namely Whitney elements (Bossavit, 1982, N´ed´elec, 1980). Its definition and complete understanding rely on the use of a geometrical point of view by involving tools from differential exterior calculus. The same point of view allow to extend the definition of Whitney elements to high-order approximations which respect the physical nature of the described fields.

BIO:
Francesca Rapetti has studied applied mathematics at the Univ. of Milan (IT) in 1995. Then, she worked two years as junior researcher at the CRS4 laboratory in Sardinia (IT), under the scientific supervision of Prof. Quarteroni. In 1997, thank to a fellowship from the European Community, she could reach the Univ. of Paris VI (FR) to start a PhD thesis under the scientific direction of Prof. Maday. In 2001, she got a permanent position as assistant professor in applied mathematics at the Univ. of Nice (FR).
Main research interests cover domain decomposition techniques for PDEs, Maxwell equations and Galilean electromagnetism, mortar element methods, differential forms and Whitney finite elements in scientific computing, h, p, and hp-finite element/spectral element methods on simplices, numerical simulations for industrial applications, numerical analysis, linear algebra and computer code development for PDEs.

HOST: Prof. Mauro Pezzè