Mat-1.3656 Seminar on
numerical analysis and computational science
Monday Oct 13, 2008, room
U356 at 14.15, Eirola & Stenberg
Mika Juntunen ja Juho Könnö, TKK/Mat
Mika Juntunen:
Analysis of finite element methods for the Brinkman problem
The Brinkman equations is a combination of Stokes and Darcy
equation. We analyze a situation where we move with a parameter from
the Stokes (or Brinkman) equation to the Darcy equation. We show the
natural
framework for the analysis of the Brinkman system. The framework is
capable of
analysing accurately even the limiting Darcy equations. We use MINI-type
elements or stabilized methods to discretize the system and show that
the method
is stable for both the Stokes and Darcy equations.
Juho Könnö:
Finite element analysis of Biot's consolidation problem
We
consider finite element approximations for the Biot consolidation model
describing fluid-filled porous materials composed of incompressible
grains. Biot's model includes both the displacement field u and the
pore pressure p. The model can be applied to a variety of situtations
involving stability of soil layers, for example nuclear waste disposal
and oil recovery. We focus on the stability and accuracy of the
time-dependent finite element solution for the problem. By eliminating
the displacement field from the equations we show the close connection
of the problem to the heat equation. This in turn allows us to apply
standard techniques for parabolic equations for the convergence
analysis of the pore pressure. We also present a practical
time-stepping scheme based on the familiar mixed finite element
formulation for the Stokes problem, thus solving both the displacement
and pore pressure fields simultaneously on each timestep. Furthermore,
closer examination and experiments show a close connection with this
particular time-stepping scheme and the stabilized version of equal
order Stokes elements, allowing the use of lower order elements.