Mat-1.3656 Seminar on
numerical analysis and computational science
Monday, Feb 6, 2012, room
U322 at 14.15, Eirola & Stenberg
Antti Niemi, King Abdullah University of Science and Technology (KAUST)
Automatically Stable Discontinuous Petrov-Galerkin Method for Some Parameter-Dependent Problems
Mixed finite element methods, where both displacements and stresses
are declared as independent unknowns, are preferred over merely
displacement-based methods in some situations. Perhaps the most repeated
reason is that the stress variables are of the most interest and mixed
methods allow their direct approximation. If a displacement formulation
is used, the stress fields must be derived from the displacement field
and it is well known that the relative accuracy of the stress field can
be much lower than that of the displacement field. Another advantage of
mixed methods is that they are often robust in parameter-dependent
problems, that is, uniformly convergent for all values of a critical
physical parameter. Typical examples of such parameter are the Poisson
ratio of an elastic material, slenderness of a structure, Peclet number
in fluid flow, etc.
In contrast to pure displacement formulations, mixed
finite element methods do not inherit stability from the continuous
variational formulation, but the stability of the discretization must be
independently verified for each choice of the finite element spaces.
The recently introduced discontinuous Petrov-Galerkin (DPG) method with
optimal test functions addresses this problem by providing means for
automatic computation of test functions that guarantee discrete
stability for any choice of trial functions. In this talk, we will
present the general variational framework of the method as well as
applications to stationary transport problems and simulation of
thin-walled structures. We will show theoretical and numerical results
with comparisons to some traditional formulations used in engineering
practice.