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 {\bf
 4th International Conference on Numerical Methods and Applications:
 NM\&A -- O(h$^4$) '98, Sofia, Bulgaria, August 19--23, 1998.\\[3ex]
 Minisymposium on the topic ``Multifield Problems''\\[1ex]
 }
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 Abstract of the proposed lecture\\[2ex]
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 {\bf
 Domain decomposition methods and far--field boundary conditions for
 two--dimensional compressible viscous flows around elastic airfoils\\[2ex]
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 %{\bf
 {\sc  C. Coclici and W.L. Wendland}\\[1ex]
 %C. Coclici, W.L. Wendland\\[1ex]
 {\sc Mathematical Institute A, University of Stuttgart}\\[3ex]
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 Domain decomposition and zonal methods are efficient
 techniques in the numerical treatment of boundary value problems in
 Computational Fluid Dynamics. Exterior boundary value problems, as for example
 the transonic flow around an elastic airfoil, permit the decomposition
 of the initially unbounded domain into a bounded computational domain
 and the corresponding far--field. Moreover,
 different physical phenomena play different r\^{o}les in corresponding regions
 and motivate a domain decomposition into respective
 model zones.\\[2ex]
 The complete system of conservation laws in the chosen
 bounded computational domain can be coupled with different, in general simpler
 conservation law models in the far--field. By solving these
 simplified far--field models,
 %(very
 %often modelled as independent boundary value problems)
 one obtains
 artificial boundary conditions for the interior
 problem. Such a domain decomposition permits to generate effective
 methods for solving the original boundary value problem, since the
 computation is done in a considerably smaller domain whereas the
 far--field boundary conditions can be realized via fast methods.
 The accuracy of the solution and the computational costs depend essentially
 on the imposed transmission conditions (far--field boundary conditions for
 the interior problem).
 \\[2ex]
 We propose two methods to construct
 artificial far--field boundary conditions for the numerical treatment
 of two--dimensional compressible, viscous flows around airfoils.
 The domain decomposition {\sl (with and without overlapping)} is used.
 Near to the given profile, the flow is modelled by the unsteady, com\-pres\-sible
 Navier--Stokes equations.  The far--field is described by simplified
 models.
 %, taking  there into consideration the physical properties of the
 %flow.
 In a first approximation, the far--field is modelled by the linearized
 Euler equations. From the corresponding integral representation of the
 far--field solution, non--local far--field boundary conditions
 for the interior problem are generated.
 In the second step, the compressible flow is considered
 as inviscid and irrotational between the Navier--Stokes region and the
 far--field; and the full potential equation is discretized there.\\[2ex]
 The coupling procedures which we propose take into account the complex nature
 of the viscous, compressible flow in the closed neighbourhood of the profile.
 The compressibility of the irrotational flow in front and on the side of the
 profile is there taken into account by the full potential equation. Finally,
 the vorticity in the wake domain is included by the linearized Euler
 equations.

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