BAIL 2006 Book of Abstracts - Institut für Numerische und ...

M. BAUSE: Aspects of SUPG/PSPG and GRAD-DIV Stabilized Finite Element
Approximation of Compressible Viscous Flow
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Aspects of SUPG/PSPG and GRAD-DIV Stabilized
Finite Element Approximation of Compressible Viscous Flow
Markus Bause
Institut für Angewandte Mathematik, Universität Erlangen-Nürnberg
Martensstr. 3, 91058 Erlangen
bause@am.uni-erlangen.de
In this contribution various aspects of a theoretical analysis and numerical study of
SUPG/PSPG and grad-div stabilized finite element approximations (cf. [3]) of steady
and unsteady compressible isothermal viscous flow are addressed. After temporal discretization
of the time variable by means of the implicit Euler method, the equations of
compressible viscous flow are solved by an iteration between a generalized Oseen problem
for the velocity and a hyperbolic transport equation for the perturbation from the mean
density (cf. [1]). Such a splitting-type approach seems attractive for computations because
it offers a reduction to simpler problems for which highly refined numerical methods either
are known or can be built from existing discretization techniques for similar equations,
and because of the guidance that can be drawn from an existence theory based on it.
In the case of steady motions of a compressible viscous gas, decribed by the equations
∇ · (ρv) = 0 , ρv · ∇v − µ∆v − (λ + µ)∇∇ · v + ∇p = ρf ,
p = kρ , v|∂Ω = 0 ,
�
Ω ρ dx = M
the iteration for solving this system reads as: Let ρ = 1 + σ. Put v0 = 0, σ0 = 0. For
given vn, σn compute vn+1, σn+1 by
(i) solving the generalized Oseen-system
∇ · vn+1 = −∇ · (σnvn) ,
(1 + σn)vn · ∇vn+1 + 1
2 ∇ · ((1 + σn)vn)vn+1 − µ∆vn+1 + ∇πn+1 = (1 + σn)f ,
vn+1|∂Ω = 0,
�
Ω πn+1 dx = 0
(ii) and, then, solving the hyperbolic transport equation
kσn+1 + (λ + 2µ)∇ · (σn+1vn+1) = πn+1 − µ∇ · vn+1 .
For the approximation of the separated Oseen problem, SUPG/PSPG and grad-div stabilized
higher order finite techniques based on LBB-stable elements are used and analyzed.
The transport equation is discretized by SUPG stabilized finite element methods. Error
estimates are provided. Numerical results for realistic steady and unsteady benchmark
problems (driven cavity flow, flow over a backward facing step and DFG-benchmark) are
given for a large scale of Reynolds numbers.
References
[1] M. Bause, J. G. Heywood, A. Novotny and M. Padula, On some approximation schemes
for steady compressible viscous flow, J. Math. Fluid Mech., 5 (2003), pp. 201–230.
[2] M. Bause, Stabilized finite element approximation of compressible viscous flow, to appear, 2006.
[3] T. Gelhard et al., Stabilized finite element schemes with LBB-stable elements for incompressible
flow, J. Comput. Appl. Math. 177 (2005), pp. 243–267.
Speaker: BAUSE, M. 72 BAIL 2006
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