Outline: Saddle Point Problems - Lehrstuhl Numerische Mathematik

Transcrição

Prof. Dr. Barbara Wohlmuth
Lehrstuhl für Numerische Mathematik
Outline: Saddle Point Problems
I. Examples for saddle point problem: Stokes problem, contact problems,
mortar method
II. Stability of saddle point problems and their discretizations
III. Iterative solvers
Kapitel III (SPPoutline)
1
Specialized Literature
• Theory on saddle point problems in general
– D. Braess: Finite Elemente. Theorie, schnelle Löser und Anwendungen in der
Elastizitaetstheorie, Springer-Lehrbuch, 2003.
– S. Brenner, R. Scott: The Mathematical Theory of Finite Element Methods,
Springer Texts in Applied Mathematics, Vol. 15, 2008
– D. Boffi, F. Brezzi et.al.: Mixed Finite Elements, Compatibility Conditions,
and Applixations, Springer, 2008
• Mortar methods
– A. Quarteroni, A. Valli: Domain Decomposition Methods for Partial
Differential Equations, Qxford Science Publications, 1999 (Chapter 2.5.1)
• Iterative solvers
– M. Benzi, G.H. Golub, J. Liesen: Numerical solution of saddle point problems,
Acta Numerica(2005), pp. 1-137
Kapitel III (SPPoutline)
2
Reminder: Quadratic Programming
The constrained quadratic minimization problem
1 T
x Ax + f T x
2
s.t. Bx = g
min
can be transformed to a linear equation system, if we introduce the Lagrange
multiplier λ.
Kapitel III (2)
T
f
u
A B
=
g
λ
B 0
3
PDEs as constrained minimization problems
Some techniques from finite dimensional minimization also work for variational
formulations. As an example, the Stokes equation is equivalent to the energy
minimization problem (see your homework sheet)
1
s.t. div u = 0
min k∇uk20 + hf , xi,
2
In this case, the pressure p works as the Lagrange multiplier:
−∆u + ∇p = f in Ω,
div u = 0 in Ω
pressure field
Streamlines: uniform
20
Example: Flow through
a channel with a ”step”.
15
10
5
0
0
1
2
4
Kapitel III (3)
0
−1
4
Contact Problems: Two body contact
Energy minimization with an inequality constraint
1
ε(V ) : Cε(v) −
fv
v∈K 2 Ω
Ω
2
2
1
1
K = {v ∈ H0,ΓD (Ω1) × H0,ΓD (Ω2) , g − [v · n] ≥ 0 on ΓC }
inf
Z
Z
g − [ v · n ] is the gap between the two deformed bodies
([ · ] is the jump of the function on ΓC )
Kapitel III (5)
5
Contact Problems in life and industry
⇒ fine geometrical resolution
nearly incompressible materials
⇒ small forming zone
thermal and plasticity effects
The contact condition yields an inequality constraint
for the energy minimization.
Kapitel III (4)
6
Mortar: Tearing and connecting
master
tearing
dual variable for interface Γ
connecting
slave
Interface conditions on Γ := ∂Ωm ∩ ∂Ωs
um = us
σn(um) = σn(us)
Kapitel III (1)
7
Mortar Methods: General Idea
−∆u = f, in Ω,
u = g on ∂Ω,
Ω = (0, 2) × (0, 1)
γ
• Decompose the geometry
Ωm
Ωs
• Discretize the pde independently on each subdomain
• Ensure global continuity in a weak sense, by integral equations:
Z
(um − us)Ψj dx = 0
γ
(these constraints yield a saddle point problem)
Kapitel III (1)
8
Mortar: Optimal Convergence in Different Situations
straight and curvilinear different meshsize ratios
Kapitel III (1)
different number of
subdomains
9
Inf-Sup Condition
pressure field
• b inf-sup stable:
20
b(v, µ)
inf sup
≥β>0
µ∈M v∈X kvkX kµkM
15
10
5
0
1
−5
0
0
2
4
• a coercive on {v ∈ X : b(v, µ) = 0, µ ∈ M }
−1
=⇒
an unstable discretization
pressure field
The saddle point problem
20
15
a(u, v) + b(v, λ) = f (v),
10
b(u, µ) = g(µ),
5
0
v∈V
µ∈M
1
0
2
0
4
−1
has a unique solution (u, µ) ∈ X × M .
a stable discretization
Kapitel III (6)
10
Checkerboard instability
Q1 × P0 is not inf-sup stable. (⇔ ∃p0 ∈ Mh : b(vh, p0) = 0, ∀vh ∈ Vh)
The checkerboard mode serves as a counter-example:
Arbitrary velocity (in each direction)
Checkerboard−mode: p
c
1
0.5
0.8
0.4
0.3
0.2
0.2
0
0
−0.5
−0.2
−0.4
−1
−0.6
1
0.5
0
y
0.2
0
0.6
0.4
−0.8
1
0.8
Arbitrary velocity field:
u
0.6
0.5
p
The pressure mode:
0.4
1
0.1
0
−0.1
−0.2
1
0.8
−1
1
0.6
0.8
0.6
0.4
0.2
x
0.2
0
y
Pn−1 Pn
0.4
0
x
R
b(vh, pc) = i=1 j=1 ev u 2(−1)i+j−1 + . . . (vertical part)
ij
These are integrals along grid lines. The integral along each line is zero.
1
0.4
0.9
0.3
0.8
0.2
0.7
0.1
0.6
i-th horizontal line
The integrand has zero mean value:
0.5
0.4
0
−0.1
0.3
−0.2
0.2
−0.3
0.1
0
Kapitel III (7)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
11
Driven Cavity: Checkerboard Modes
Q1P0-Discretization
Pressure along y = 0.22
u = (1, 0)T
20 × 20 elements
0.2
Ω = (0, 1)
2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
−0.05
−0.05
−0.1
−0.1
−0.15
−0.15
−0.2
Kapitel III (7)
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
40 × 40 elements
−0.2
1
12
Is there any remedy for the inf-sup stability
The special structure of the inf-sup condition
b(vh, ph)
inf sup
≥β>0
ph ∈Mh v ∈V kvh kkph k
h
h
shows us two possibilities, that might improve the inf-sup stability
• Enrich the velocity space Vh
• Reduce the pressure space Mh.
Further possibilities are to generalize the saddle point formulation, e.g. by
• penalization,
• stabilization or by
• constructing a divergence-free velocity space (for a divergence-free Stokes)
Kapitel III (8)
13
Reducing the pressure space: Makro elements
The instability of the checkerboard mode shows us, which elements we have to
remove from the pressure space:
+ - +
Remaining basis functions on a makro element (four elements grouped together):
+ +
+ +
+ +
- -
+ + -
Uniform stability (see e.g. Braess)
Kapitel III (9)
14
Enhancing the velocity space: Taylor Hood, MINI
The Taylor-Hood element has two versions. One enlarges the polynomial degree
of the velocity space, the other one defines the velocity space on a finer mesh.
Vh
Mh
The MINI element adds bubble-functions to the velocity space to ensure inf-sup
stability. Compared to the Taylor-Hood elements this yields a smaller dimension
and an easy solution by static condensation.
Vh
Mh
• cubic bubble function.
Kapitel III (10)
15
Penalization
Instead of demanding div v = 0 directly, we can introduce it as a penalty term:
1
min
2
Z
(∇v)2 + ε−1 (div v)2 − 2f v dx
Using p = ε div u yields the weak formulation
a(u, v) + b(v, p) = f (v)
v ∈ H01(Ω)d,
b(u, q) − ε(p, q)0,Ω = 0 q ∈ L2,0(Ω).
The discretized system is a linear system of the form
T
f
u
A B
=
0
p
B −εC
For ε → 0, we have convergence to our solution.
Kapitel III (11)
16
Stabilization
Another stabilization is based on adding a least-squares term (based on the strong
formulation) to the minimization:
k − ∆u + ∇p − f k20,Ω
For linear velocity elements, the variational equation yields the simple form to find
(uh, ph) ∈ Vh × Qh ⊂ H 1(Ω)d × H 1(Ω), such that
a(uh, vh ) + b(vh, ph) = f (vh), v ∈ Vh
X
X
2
h2T (∇qh, fh)0,T ,
hT (∇qh, ∇ph)0,T = −α
b(uh, qh) − α
T ∈Th
qh ∈ Qh.
T ∈Th
More details, e.g. in
Donea, J., Huerta, A. Finite Element Methods for Flow Problems, Wiley, 2003
(Chapter 6.5.8)
Kapitel III (13)
17
Divergence-free non-conforming Crouzeix-Raviart element
The Crouzeix-Raviart element is divergence-free by construction. Hence, the
solution can be computed without the construction of a pressure space (which can
still be constructed as a post-process to compute the pressure).
1
d
The downside is, that the elements are nonconforming, due to Xnc
h 6⊂ H (Ω) .
2
1
1
0
0.8
−1
0.6
−2
1
0.4
0.8
0.6
0.2
0.4
0.2
0
Kapitel III (12)
0
18
H(div, Ω)-finite elements
Mixed formulation of the Poisson problem: (σ, u) ∈ H(div, Ω) × L2(Ω)
(σ, τ )0,Ω + (div τ, u)0,Ω = 0 τ ∈ H(div, Ω),
(divσ, v)0,Ω = −(f, v)0,Ω
Example: Raviart–Thomas–elements
RT0 in 2D
RT1 in 2D
v ∈ L2(Ω).
RT0 in 3D
Ansatz for RT0:
 
 
a
x
φ(x, y, z) =  b  + d y 
c
z
Kapitel III (fe15eng)
Further examples for H(div, Ω)-elements
- Brezzi–Douglas–Marini
- Brezzi–Douglas–Fortin–Marini
19

Outline: Saddle Point Problems - Lehrstuhl Numerische Mathematik

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