CRM-McGill Applied Math Seminar
Tuesday Mar 25, 2008, 3:35pm
Burnside Room 1205

Speaker: Benjamin Stamm
Institute of analysis and scientific computing,
Ècole Polytechnique Fédérale de Lausanne,

Title:
Bubble stabilized discontinuous Galerkin method for elliptic, parabolic and
Stokes' problems

Abstract:
In the first part of the talk we present a symmetric DG method using a
piecewise affine approximation space, enriched by nonconforming quadratic
bubbles. This symmetric scheme is stable and optimally convergent without
interior penalty term. This unstabilized approach has some interesting
properties: the method enjoys enhanced local mass conservation unperturbed
by any numerical parameters and the system is adjoint consistent allowing
for optimal convergence of the error in the $L^2$--norm. On the other hand,
the system resulting from the proposed discretisation is not positive
definite, leading to a more complex analysis. Indeed standard coercivity
arguments fail and an argument involving an inf-sup condition must be
applied also for the classical Poisson problem.

In the second part of this talk we will show how the proposed method can be
extended to more complex problems in fluid mechanics with special focus on
the heat equation and Stokes' problem.

For the parabolic model problem we show that in spite of the presence of
negative eigenvalues a standard backward Euler time discretization scheme
leads to a stable and optimally convergent method. Stability holds under a
non restrictive inverse parabolic CFL--condition, namely that $h^2/\delta t$
is small enough where we denote by $h$ the spatial and by $\delta t$ the
temporal discretization parameter.

For Stokes' problem on the other hand we investigate what pressure spaces
may be used in order for the problem to be uniformly wellposed. We propose
to combine the bubble enriched space for the velocities with the space of
discontinuous piecewise constant functions or the space of continuous affine
approximations for the pressure. We will discuss how these choices of
pressure spaces relates to classical methods indicating in each case how the
inf-sup condition may be proved. Moreover we discuss the approximation
properties of both variants and review the conservation properties of the
resulting method.

Some numerical examples illustrating the theory will be given for the
model-problems discussed.