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Nicolas Neuss
| Speaker: |
Nicolas Neuss (University of Karlsruhe)
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| Date: |
Friday March 13, 2009
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| Title: |
Numerical
approximation of boundary layers for rough boundaries
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Abstract
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In
physical problems, interesting phenomena often occur at boundaries or
interfaces between different media. Often these phenomena are
complicated due to the nature of the process or due to the intricate
geometry of the interface. Therefore, they are usually described
by effective boundary or interface laws.
In this talk we will discuss some instructive cases in a quasi-periodic
setting, where the constants in those effective conditions can be
calculated from the microscopic setting.
First, we consider the Poisson problem on a domain with an oscillating
boundary. We formulate an effective equation posed on an
approximating domain with smooth boundary. The parameters in this
effective equation can be computed solving auxiliary problems taking
into account the fine-scale structure, see [NRM].
Similarly, solving the Navier-Stokes equation in a domain with rough
boundary can be replaced by solving it on a smoothly bounded domain
with a Navier boundary condition. Related to this is the
Beavers-Joseph problem, where one can derive an effective interface law
for a rough interface between free flow and porous medium flow, see
[JMN].
Finally, the same idea can be used to obtain effective boundary
conditions for numerical approximations of arbitrary domains.
This allows us to obtain a second order approximation error in the
solution although the domain approximation is only of first order.
Literature:
[JMN] W. Jaeger, A. Mikelic, N. Neuss: Asymptotic Analysis of the
Laminar Viscous Flow Over a Porous Bed}. SIAM J. Sci. Comp. 22,6,
pp. 2006-2028 (2001).
[NRM] N. Neuss, M. Neuss-Radu, A. Mikelic: Effective Laws for the
Poisson Equation on Domains with Curved Oscillating Boundaries.
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