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

G. MATTHIES, L. TOBISKA: Mass conservation of finite element methods for coupled
flow-transport problems
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Mass conservation of finite element methods for coupled flow-transport
problems
Gunar Matthies 1 and Lutz Tobiska 2
1 Fakultät für Mathematik, Ruhr-Universität Bochum
2 Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg
We consider a coupled flow-transport problem in a bounded domain Ω ⊂ R d , d = 2,3. The
system is described by the instationary, incompressible Navier–Stokes equations
and the time-dependent transport equation
ut − ν△u + (u · ∇)u + ∇p = f in Ω × (0,T],
div u = 0 in Ω × (0,T],
u = ub on ∂Ω × (0,T],
u(0) = u 0
in Ω,
ct − ε△c + u · ∇c = g in Ω × (0,T],
(cu − ε∇c) · n = cI u · n on Γ− × (0,T],
ε∇c · n = 0 on Γ+ × (0,T],
c(0) = c 0
in Ω.
Here, u and p denote the velocity and the pressure of the fluid, respectively, ν and ε are small
positive numbers, c is the concentration, cI the concentration at the inflow boundary Γ− :=
{x ∈ ∂Ω : u · n < 0}, and Γ+ := ∂Ω \ Γ−. We assume that ub is the restriction of a divergence
free function onto the boundary ∂Ω. Note that the velocity u from the Navier–Stokes equations
enters the transport equation as a convection field.
Due to incompressibility constraint, the weak solution c of (2) satisfies the global mass
conservation property
� � � �
d
cdx + cIu · ndγ + cu · ndγ = g dx. (3)
dt Ω Γ−
Γ+
Ω
It is well-known [1], that the finite element solutions satisfy in general the incompressibility
constraint and the global mass conservation property (3) only approximately.
Different discretisation methods [2, 3, 4] for the instationary, incompressible Navier–Stokes
equations and stabilised schemes for the transport problem like SDFEM will be studied numerically.
References
[1] C. Dawson, S. Sun, M. F. Wheeler, Compatible algorithms for coupled flow and transport.
Comput. Methods Appl. Mech. Engrg., 193, 2565–2580 (2004).
[2] G. Matthies, L. Tobiska, The inf-sup condition for the mapped Qk/P disc
k−1 element in arbitrary
space dimensions. Computing, 69, 119–139 (2002).
[3] G. Matthies, L. Tobiska, Inf-sup stable non-conforming finite elements of arbitrary order
on triangles. Numer. Math., 102, 293–309 (2005).
[4] G. Matthies, Inf-sup stable nonconforming finite elements of higher order on quadrilaterals
and hexahedra. Bericht Nr. 373, Fakultät für Mathematik, Ruhr-Universität Bochum
(2005).
Speaker: MATTHIES, G. 115 BAIL 2006
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