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DiscretizationDiscretizationand Solution of and Solution of ConvectionConvection--Diffusion ProblemsDiffusion Problems
Howard ElmanUniversity of Maryland
1
Overview
1. The convection-diffusion equationIntroduction and examples
2. Discretization strategiesFinite element methodsInadequacy of Galerkin methodsStabilization: streamline diffusion methods
3. Iterative solution algorithmsKrylov subspace methodsSplitting methodsMultigrid
2
The Convection-Diffusion Equation
3,2,1 ,in 2 =⊂Ω=∇⋅+∇− dfuwu dRε
Boundary conditions:
NN
DD
gnu
gu
Ω∂=∂∂
Ω∂=
on
on
D
N
Ω∂Ω∂Ω
Inflow boundary: ∂Ω+ = x∈∂Ω | w· n > 0
Characteristic boundary:∂Ω0 = x∈∂Ω | w· n = 0
Outflow boundary:∂Ω- = x∈∂Ω | w· n < 0
3
The Convection-Diffusion Equationfuwu =∇⋅+∇− 2ε
Challenging / interesting case: ε 0
Reduced problem: w·∇ u = f , hyperbolic
Streamlines: parameterized curves c(s) in Ω s.t. c(s)has tangent vector w(c(s)) on c
fscudsdwu ==⋅∇⇒ ))(()(
Solutionu to reduced problem = solution to ODE
If u(s0)∈ inflow boundary ∂Ω+, and u(s1)∈∂Ω, say outflow ∂Ω− ,then boundary values are determined by ODE
4
Consequence 2 fuwu =∇⋅+∇−ε
For small ε, solution to convection-diffusion equation often hasboundary layers, steep gradients near parts of ∂Ω.
Αlso: discontinuities at inflow propagate into Ω along streamlines
Simple (1D) example of first phenomenon:
1near except 11
)(
/1
/)1)(/1(
=≈
−−−
=−
−−
xxeee
xxux
xε
εε
Solution
0)1( ,0)0(
,)1,0(on 1'''
===+−
uu
uuε
Solution to reduced equation
5
Additional Consequence
These layers (steep gradients) are difficultto resolve with discretization
6
Conventions of Notation
fLuwWLu
=∇⋅
+∇− εε
2
***2
In normalized variables:
,εWLP ≡ Peclet number, characterizes relative contributions
of convection and diffusion
L = characteristic length scale in boundarye.g. length of inflow boundary
= x/L in normalized domain
W = normalization for velocity (wind) w,e.g. w = W w*, where ||w*||=1
x
7
Reference Problems
−
−= −
−−
ε
ε
/2
/)1(
11),(
eexyxu
y
2 fuwu =∇⋅+∇−ε
1. w=(0,1)Dirichlet b.c.analytic solution
2. w=(0,(1+(x+1)2/4)Neumann b.c. at outflowcharacteristic boundary layers
8
Reference Problems 2 fuwu =∇⋅+∇−ε
3. w: 30o left of verticalinterior layer from
discontinuous b.c.downstream boundary layer
4. w=recirculating(2y(1-x2),-2x(1-y2)characteristic boundary layersdiscontinuous b.c.
9
Weak Formulation 2 fuwu =∇⋅+∇−ε
on 0 | )(
on | )(
)(
),( allfor s.t. )( Find
1
1
11
0
0
DE
DDE
N
EE
vvH
gvvH
vgfvvuwvu
HvHu
N
Ω∂==Ω
Ω∂==Ω
+=∇⋅+∇⋅∇
Ω∈Ω∈
∫∫∫Ω∂ΩΩ
ε
Shorthand notation: a(u,v) =l(v) for all v
Can show:
a(u,u) ≥ ε ||∇u||2 = ε∫Ω∇ u ·∇ u
a(u,v) ≤ (ε+||w||∞ L) ||∇u|| ||∇v||
l(v) ≤ C ||∇v||
Lax-Milgram lemmaexistence and uniquenessof solution
10
Approximation by Finite Elements
a(uh,vh) =l(vh) for all vh
Typically: finite element spaces are defined by low-orderbasis functions, e.g. — linear or quadratic functions on triangles— bilinear or biquadratic functions on quadrilaterals
∫∫∫∫ Ω∂ΩΩΩ+=∇⋅+∇⋅∇
∈∈
⊂⊂
NNhhhhhh
hh
hEh
Eh
EhE
gvvfvuwuu
SvSu
HSHS
)(
, allfor such that find
, , ldimensiona finiteGiven
0
10
1
0
ε
11
What happens in such cases?
Problem 1, accurate
Problem 2, accurate
Problem 1, inaccurate
Problem 2, inaccurate
12
Explanations
small is if large ,1 1
,)(inf)(
w
h
εε
εPWL
vuuu hSvw
h hE
+=+=Γ
−∇Γ≤−∇∈
1. Error analysis: discrete solution is quasi-optimal:
2. Mesh Peclet number: 2
2 εWh
LPhPh ==
If Ph>1, then — there are oscillations in the discrete solution— these become pronounced if mesh does not resolve layers— oscillations propagate into regions where solution is smooth— problem is most severe for exponential boundary layers
13
Revisit two examples
Problem 1, exponential layer, width ~ ε
Problem 2, characteristic layer, width ~ ε1/2
14
Fix: The Streamline Diffusion Method
Petrov-Galerkin method: change the test functions
Galerkin: a(uh,vh) =l(vh) for all vh
Petrov-Galerkin: a(uh,vh+δ w·∇ vh) =l(vh+δ w·∇ vh) for all vh
δ is a parameter
Result: asd(uh,vh) =lsd(vh)
)()()(
)()(
))(()(),(
2
∫∫∫
∑ ∫∫∫∫
Ω∂ΩΩ
∆
ΩΩΩ
∇⋅++∇⋅+=
∇⋅∇−
∇⋅∇⋅+∇⋅+∇⋅∇=
N
k
Nhhhhh
hk h
hhhhhhhhsd
gvwvvwfvfvl
vwu
vwuwvuwuuvua
δδ
δε
δε
Streamline diffusion term
0 for linear/bilinear
15
The Streamline Diffusion Method Explained
Augment finite element space:
Bh: bubble functions, with support local to element
hBSS += hhˆ
We could pose the problem on the augmented space:find uh in s.t. a(uh,vh) =l(vh) for all vh in
Then: decouple unknowns associated with bubble functionsfrom system new problem on original grid
Principle: augmented space places basis functions inlayers not resolved by the grid
hS
hS hS
16
The Streamline Diffusion Method Explained
Under appropriate assumptions: this new problem is
)()(
))((
)(),(
∑ ∫∫
∑ ∫
∫∫
∇⋅+=
∇⋅∇⋅+
∇⋅+∇⋅∇=
∆Ω
∆ ∆
ΩΩ
k hkhh
hhk
hhhhhhsd
vwfvfvl
vwuw
vuwuuvua
k
k k
δ
δ
ε
asd(uh,vh) =lsd(vh)
k determined from elimination of bubble functionsStreamline diffusion
17
Compare Galerkin and Streamline Diffusion
Middle: bilinear elements, Galerkin,32× 32 grid
Top: accurate solution, ε=1/200
Bottom: bilinear elements, streamline diffusion,32× 32 grid
18
Error Bounds
)(inf)(h
hSvw
h vuuu hE
−∇Γ≤−∇∈ε
For Galerkin: as noted earlier, quasi-optimality:
More careful analysis: for linear/bilinear elements,
)( 2uDChuu h ≤−∇ Large in exponential boundary layers for small ε
For streamline diffusion: use norm
( ) 1/222
vwvvsd
∇⋅+∇≡ δε
22/3 uDChuusdh ≤−
Then
19
These bounds do not tell the whole story
4.98e-74.98e-72.392.3964×64
(1)
1.11e-55.30e-23.233.8132×32
(2)
1.64e-51.484.014.9116×16
(4)
8.16e-73.254.345.628×8(8)
Str.Diff.Ω∗
GalerkinΩ*
Str.Diff.Ω
GalerkinΩ
Grid (Ph )
For one example (Problem 1, ε=1/64), compareerrors ||∇(u-uh)|| on Ω and Ω* = (-1,1)×(-1,3/4) (to exclude boundary layer)
20
Choice of parameter δδδδ
∑ ∫
∫∫
∆ ∆
ΩΩ
∇⋅∇⋅+
∇⋅+∇⋅∇=
k khhk
hhhhhhsd
vwuw
vuwuuvua
))((
)(),(
δ
εMade element-wise:
( )
≤
>−=
1 if 0
1 if /11||2
kh
kh
kh
k
k
k
P
PPw
hδ
21
Matrix Properties
nila
u
SS
jjij
j jjh
hE
nnnj
hnjj
,...,2,1 ),(u),(
such that u find : becomes
problem , u isfunction element Finite
for by extended ,for basis aGiven
j
j
101
==
=
∑
∑∂+
+=
ϕϕϕ
ϕϕϕ
or asd
Leads to matrix equation Fu=f,
F=ε A + N (+ S)
22
Matrix Properties
Matrix equation Fu=f, F=ε A + N (+ S)
A=[aij], aij=∫Ω ∇φj·∇φi, , discrete Laplacian, symmetric positive-definite
N=[nij], nij=∫Ω (w·∇φj)φi , discrete convection operator,skew-symmetric (N=-NT)
S=[sij], sij=∫Ω (w·∇φj)(w·∇φi) , discrete streamlineupwinding operator, positive semi-definite
23
End of Part I
Next: how to solve Fu=f ?
24
Iterative Solution Algorithms: Krylov Subspace Methods
System Fu=f
• F is a nonsymmetric matrix, so an appropriate Krylov subspacemethod is needed
• Examples:• GMRES• GMRES(k) restarted• BiCGSTAB• BiCGSTAB(l)
• Our choices: • Full GMRES for optimal algorithm, or • BiCGSTAB(2) for suboptimal
25
Properties of Krylov Subspace Methods
Drawback of GMRES: work & storage requirements at step k
are proportional to kN
BiCGSTAB: Fixed cost per step, independent of k
Drawback: No convergence analysis
Variant: BiCGSTAB(l), more robust for complex eigenvalues,
somewhat higher cost per step (l=2), but still fixed
26
Convergence of GMRES
GMRES: Starting with u0, with residual r0=f-Fu0, computesuk ∈ span r0, Fr0, …,Fk-1r0
for which rk = f-Fuk satisfies
||rk|| = minpk(0)=1 || pk(F)r0||.
Consequence:
Theorem: For diagonalizable F=VΛV-1,
||rk|| ≤ ||V|| ||V-1|| minpk(0)=1 maxλ∈σ(F)|pk(λ)| ||r0||.
(||rk||/||r0||)1/k ≤ (||V|| ||V-1||)1/k (minpk(0)=1maxλ∈σ(F)|pk(λ)|)1/k
ρLoosely speaking: residual is reduced by factor of at each step
Want eigenvalues to lie in compact setρ
27
Convergence of GMRES
)(11
ˆ 1
2
22
FF RQad
daa −=≤−+−+≈ ρρ
0 1
.
.1+d
1+a
Size of convergence factor ρ
28
Key for Fast Convergence: PreconditioningSplitting operators
SeekQF≈ F such that• the approximation is good, and• it is inexpensive to apply the action of Q-1 to a vector
Splitting: F=QF-RF stationary iteration
Error ek = u-uk satisfies
)(11 fuRQu kFFk += −
+
( ) )()(/
)(
)(
))(()(
1/11/1
0
01
01
111
FQIFQIee
eFQIe
eFQIe
uuFQIuuRQuu
F
kkF
k
k
kFk
kFk
kFkFFk
−−
−
−
−−+
−≈−≤
−≤
−=
⇒−−=−=−
ρ
29
Preconditioning / Splitting operators
Thus: want to be as small as possibleEquivalently: eigenvalues of
= eigenvalues of
)( 1FQI F−−ρ
FQF1−
1−FFQ
as close to 1 as possible
0 1
)( 1FF RQ−ρ
This is similar to the requirement for rapid convergence of GMRES
Solve uQufuFQfQFuQ FFFF ˆ ,ˆor 1111 −−−− ===
30
Examples of splitting operators
=FQ lower triangle of AGauss-Seidel=FQ block lower triangle of ALine Gauss-Seidel
FFF ULQ =Symmetric versionsIncomplete LU factorization
FQLUF =≈
Comments:
• All depend on ordering of underlying grid
• Symmetric versions (symmetric GS, ILU) take some account of underlying flow
• Line/block versions can handle irregular grids
31
Convergence Analysis (Parter & Steuerwalt)
Seek maximal eigenvalue of uuRQ FF λ=−1 or uRuQ FF = λSubtract uRF λ from both sides
uRhuRuRQ FFFFh
)()( 22
11
=
=− −−λ
λλ
λ
uRhFu Fh )( 2µ=
Suggests relation to uu µ= L
L uwuu ∇⋅+∇−= 2εR to be determined
R
For many examples of splittings:)( ),,(),(2 xrrvurvuRh hhF =≈operator"tion multiplicaweak " a is 2
FRhand
(2) of eigenvalue minimal )0( =→ µµh
(1)
(2)
(defines R)
32
are known:
+
+=⇒
+
++=
=
222)0(
22222
2/2/),(
222
22)(
)sin()sin( 21
εεπεµ
εεπεµ
ππ
yx
yxjk
ywxwkj
wwr
wwkj
r
ykexjeu
Consequence: 2)0(1 1)( hRQ FF µρ −=−
For model problems:(i) r is constant (will demonstrate in a moment)(ii) on square domains, eigenvalues, eigenvectors of
uu µ= L R
To find r: consider centered finite differences
+−2hwxε
−−2hwxε
−−
2hwyε
+−
2hwyε
ε4
For horizontal line Jacobi splitting, = block tridiagonal:FQ
)(222
][ 21,1, hOuu
hwu
hwuR ijji
yji
yijF +=
++
−= −+ εεε
,2ε=⇒ r
Key point: convection terms lead to smaller convergence factors
2
222
222
211 h
ww yx
+
+−= εεπρ
34
Comments / Extensions
• Similar results obtained from matrix/Fourier analysis
• Young:
• “Multi-line” (k-line) splittings r =
• Can extend to other splittings via matrix comparison theorems
(Varga-Woźnicki):
k/2ε
2operator) Jacobi (lineoperator) Seidel-Gauss (line ρρ =
)()( 1-112
-12
11
12 RQRQQQ ρρ ≤⇒≥ −−
35
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
6
5
4
3
2
1
Limitations of analysis above: It does not discriminateamong different orderings
Natural ordering of gridLeft-to-right, bottom-to-topPlus resulting matrix structure:
Horizontal line red-black Ordering and matrix structure:
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
6
3
5
2
4
1
Young theory: spectral radii (Jacobi or Gauss-Seidel) independent of ordering
Performance of GS: depends on ordering
36
Example: Problem 1 60Pelements,linear piecewise ,)1,0( 22 ==+∇− fuu yε
Four solution strategies: line Gauss-Seidel iteration withnatural line ordering, following the flow (bottom-to-top)natural line ordering, against the flow (top-to-bottom)red-black line ordering, with the flowred-black line ordering, against the flow
10-5
10-4
10-3
10-2
10-1
100
101
102
0 10 20 30 40 50 60
*o
*o*o
k
||e^(
k)||
Naturalwith flow
Red-black
Naturalagainst flow
With
Against
37
Ordering effects
⇒=−= −0
1 )( satisfies Error eRQeuue kFFkkk
01
01 )()( eRQeRQe k
FFk
FFk−− ≤=⇒
“Classical” analysis only provides insight in asymptotic sense: 1
)(lim
/11
=−
∞→ ρ
kkFF
k
RQ
E. & Chernesky:bounds for for 1D problems
kFF RQ )( 1−
38
Practical consequencesFor nonconstant flows: inherent latencies if sweeps don't follow flowPossible fixes:• flow-directed orderings (Bey & Wittum, Kellogg, Hackbusch, Xu)• iterations based on multi-directional sweeps2D version:
Speeds convergence when recirculations are present
)(
)(
)(
)(
4/31
44/31k
2/11
32/13/4k
4/11
24/11/2k
111/4k
+−
++
+−
++
+−
++
−+
−+=
−+=
−+=
−+=
kk
kk
kk
kk
FufQuu
FufQuu
FufQuu
FufQuu
Contours of stream function
39
Summarizing with an experiment:
Eigenvalues of line-GSpreconditioned operator,vertical flow, P=40, h=1/32
Asymptotic convergence rateis faster with Krylov acceleration
However: does not overcome latencies
40
MultigridFlow-following methods are effective for convection-dominatedproblems:
But: ultimately, solvers discussed above are mesh dependent
GMRES performance for Problem 4
41
Multigrid
V-cycle multigrid:
end
iteration)next for (update
h)(postsmoot )( steps, for
update) and correction (prolong ˆ
ˆˆ problem coarse tosystem multigridapply
residual)(restrict )(r
)(presmooth )( steps, for
econvergenc until 0for
Choose
1
11
2
T
11
0
ii
FiFi
ii
h
i
FiFi
uu
fQuFQIum
ePuu
reF
FufP
fQuFQIuk
i
u
←+−←
+←=
−=
+−←=
+
−−
−−
42
Bottom Line: Performance
43214321Grid
23332533128× 128
3333243364×64
5433243332× 32
6633333416× 16
Example Example
ε=1/25 ε=1/200
Multigrid iterations for ||rk||/||r0||<10-6
43
For this to happen:
)( :Smoothing 1. 11 fQuFQIu FiFi−− +−←
Two things have to be done correctly
reF h ˆˆ :solve grid Coarse 2. 2 =
Smoother must take underlying flow into account
For results above for Problem 4 (recirculating wind): smoother is 4-directional
Coarse grid operator must be stable
Even if fine grid is “fine enough,” coarse grid operators should include streamline diffusion
44
Example / effect of smoother
After one four-directionalGauss-Seidel step
After four one-directionalGauss-Seidel steps
45
Concluding Remarks
• Discretization requires stabilization for convection-dominatedproblems
• The best solution algorithms combine • general techniques of iterative methods • splitting strategies coupled to the underlying physics• stabilization when needed
46
References
• J. J. H. Miller, E. O’Riordan and G. I. Shishkin,Fitted Numerical Methods for Singularly Perturbed Problems,World Scientific, 1995.
• K. W. Morton, Numerical Solution of Convection-DiffusionProblems, Chapman & Hall, 1996.
• H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer,1996.
• H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elementsand Fast Iterative Solvers, Oxford University Press, 2005.
