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Instructor: Malgorzata Peszynska, mpesz@math.oregonstate.edu.
This course will introduce the stationary Stokes PDE
system for viscous incompressible flow, its analysis and
its finite element approximation. The model has the saddle
point structure -\mu \Delta u + \nabla p=f, \nabla \cdot u
= g. After variational formulation and numerical
discretization, the model reads AU + B^T P =F, B U=G which
is indefinite. Unlike many other elliptic PDEs which
feature coercivity in Hilbert spaces (and positive
definite linear systems), the Stokes system exploits
functional analysis setting with Banach-Necas-Babuska
conditions which derive from the Open Mapping and Closed
Range theorems. The main theoretical tool is the
generalization of coercivity to the inf-sup condition
(also known as the LBB or Ladyzhenskaya-Babuska-Brezzi);
these are verified in the continuous and discrete settings
independently. In particular, the well-known MAC scheme
will be interpreted as a lowest order mixed finite element
approximation; the stability and accuracy of other CFD
(Computational Fluid Dynamics) algorithms will be also
discussed. As time permits, we will also explore time
dependent and nonlinear (Navier-Stokes) problems, other
saddle point problems with their Lagrangians, as well as
appropriate iterative linear solvers.
Required: solid background in differential equations, real variables,
linear algebra, and experience with numerical methods. This class can
be taken independently of any other courses MTH 654-5-6/9.
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