Spectral stochastic finite elementsfor two nonlinear problems
Beďrich Soused́ık1, Howard C. Elman2
1 University of Maryland, Baltimore County2 University of Maryland, College Park
supported byDepartment of Energy (DE-SC0009301)
National Science Foundation (DMS1418754 and DMS1521563)
SISSA, QUIET workshop, July 19, 2017
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 1 / 35
I.) Inverse subspace iteration for the spectral stochastic finite elements
We are interested in a solution of
A (x , ξ) us (x , ξ) = λs (ξ) us (x , ξ) ,
where ξ = {ξi}Ni=1 is a set of i.i.d. random variables, and
A =M′∑`=0
A`ψ` (ξ) , (SPD matrix operator).
We search
λs =M∑k=0
λskψk (ξ) , us =
M∑k=0
uskψk (ξ) ,
where {ψ` (ξ)} is a set of orthonormal polynomials, 〈ψjψk〉 = 0 or 1,which is referred to as the gPC basis. We will also use the notation
c`jk = 〈ψ`ψjψk〉 = E [ψ`ψjψk ] .
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 2 / 35
Monte Carlo and stochastic collocation
Both approaches are based on sampling,
A(ξ(q))us(ξ(q))
= λs(ξ(q))us(ξ(q)), s = 1, . . . , ns .
Monte Carlo uses ensemble average
λsk =1
NMC
NMC∑q=1
λs(ξ(q))ψk
(ξ(q)), usk =
1
NMC
NMC∑q=1
us(ξ(q))ψk
(ξ(q)).
Stochastic collocation uses quadrature
λsk =
Nq∑q=1
λs(ξ(q))ψk
(ξ(q))w (q), usk =
Nq∑q=1
us(ξ(q))ψk
(ξ(q))w (q),
where ξ(q) are the quadrature points, and w (q) are quadrature weights.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 3 / 35
Stochastic Galerkin method
The stochastic Galerkin method is based on the projection
〈Aus , ψk〉 = 〈λsus , ψk〉 , k = 0, . . . ,M, s = 1, . . . , ns .
which yields a nonlinear algebraic system
M∑j=0
M′∑`=0
c`jkA`usj =
M∑j=0
M∑i=0
cijkλsi u
sj , k = 0, . . . ,M, s = 1, . . . , ns .
In order to find interior eigenvalues we use deflation via
Ã0 = A0 +
nd∑d=1
[C − λd0
] (ud0
)(ud0
)T , Ã` = A`, ` = 1, . . . ,M
′,
where(λd0 , u
d0
), d = 1, . . . , nd , are the zeroth order coefficients, i.e., the
mean value, of the eigenpairs to be deflated, and C is a constant.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 4 / 35
Algorithm: Stochastic inverse subspace iteration
Algorithm
Find ne eigenpairs of the deterministic mean value problem
A0 U = U Λ, U =[u1, . . . , une
], Λ = diag
(λ
1, . . . , λ
ne),
1 choose ns eigenvalues of interest with indices e = {e1, . . . , ens}2 set up matrices A`, ` = 0, . . . ,M ′, either as A` = A`,
or A` = Ã` using deflation3 initialize
u1,(0)0 = u
e1 , u2,(0)0 = u
e2 , . . . une ,(0)0 = u
ens ,
us,(0)i = 0, s = 1, . . . , ns , i = 1, . . . ,M.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 5 / 35
Algorithm: Stochastic inverse subspace iteration
Algorithm (cont’d)
for it = 0, 1, 2, . . .
1 Solve the stochastic Galerkin system for vs,(it)j , j = 0, . . . ,M:
M∑j=0
M′∑`=0
c`jkA`vs,(it)j = u
s,(it)k , k = 0, . . . ,M, s = 1, . . . , ns .
2 If ns = 1, normalize using the quadrature rule, or else if ns > 1orthogonalize using the stochastic modified Gram-Schmidt process.
3 Check convergence.
endUse the stoch. Rayleigh quotient to compute the eigenvalue expansions.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 6 / 35
Matrix-vector multiplication
Matrix-vector product corresponds to evaluation of the projection
vk = 〈v , ψk〉 = 〈Au, ψk〉 , k = 0, . . . ,M.
In more detail,〈M∑i=0
viψi (ξ) , ψk
〉=
〈(M′∑`=0
A`ψ` (ξ)
) M∑j=0
ujψj (ξ)
, ψk〉,
so the coefficients are
vk =M∑j=0
M′∑`=0
c`jkA`uj , k = 0, . . . ,M.
The use of this computation for the Rayleigh quotient is described next.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 7 / 35
Stochastic Rayleigh Quotient
The stochastic Rayleigh quotient defines the coefficients of a stochasticexpansion of the eigenvalue defined via a projection
λk = 〈λ, ψk〉 =〈uT v , ψk
〉, k = 0, . . . ,M,
where the coeff’s of v are computed using the matrix-vector multiplication.In more detail,〈
M∑i=0
λiψi (ξ) , ψk
〉=
〈(M∑i=0
uiψi (ξ)
)T M∑j=0
vjψj (ξ)
, ψk〉,
so the coefficients are
λk =M∑j=0
M∑i=0
cijk
〈uTi vj
〉R, k = 0, . . . ,M,
where 〈·, ·〉R is the inner product on Euclidean Ndof -dimensional space.Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 8 / 35
Normalization and the Gram-Schmidt process
The coeff’s of a normalized vector are computed from the coeff’s v1k as
u1k =
Nq∑q=1
v1(ξ(q))∥∥v1 (ξ(q))∥∥
2
ψk
(ξ(q))w (q), k = 0, . . . ,M.
The orthonormalization is achieved by a stochastic version of the modifiedGram-Schmidt algorithm [Meidani and Ghanem (2014)], which is based on
us = v s −s−1∑t=1
〈v s , ut〉R〈ut , ut〉R
ut , s = 2, . . . , ns ,
noting that 〈·, ·〉R can be again evaluated at the collocation points.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 9 / 35
Error assessment
We do not have the coefficients of the expansion of the true residual
r sk = 〈Aus − λsus , ψk〉 , k = 0, . . . ,M, s = 1, . . . , ns .
Instead, we can use a residual indicator
r̃s,(it)k =
M∑j=0
M′∑`=0
c`jkA`us,(it)j −
M∑j=0
M∑i=0
cijkλs,(it)i u
s,(it)j .
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 10 / 35
Error assessment (cont’d)
In computations, for each pair s = 1, . . . , ns , we can monitor:
1 the norms of the expected value of the indicator and its variance,
εs,(it)0 =
∥∥∥r̃ s,(it)0 ∥∥∥2, ε
s,(it)σ2
=
∥∥∥∥∥M∑k=1
(r̃s,(it)k
)2∥∥∥∥∥2
.
2 the difference of the coefficients in two consecutive iterations.
us,(it)∆ =
∥∥∥∥∥∥∥ u
s,(it)0...
us,(it)M
− u
s,(it−1)0
...
us,(it−1)M
∥∥∥∥∥∥∥
2
.
3 In addition, we can sample the true residual using Monte Carlo,
r s(ξi)
= A(ξi)us(ξi)− λs
(ξi)us(ξi), i = 1, . . .NMC .
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 11 / 35
Error assessment (cont’d)
We also report:
1 pdf estimates of the normalized `2-norms of the true residual
εsr(ξi)
=
∥∥r s (ξi)∥∥2
‖A (ξi )‖2, i = 1, . . .NMC , s = 1, . . . , ns .
2 pdf estimates of the `2-norm of the relative eigenvector approx. errors
εsu
(ξ(i))
=
∥∥us (ξ(i))− usMC (ξ(i))∥∥2∥∥usMC (ξ(i))∥∥2 , i = 1, . . . ,NMC , s = 1, . . . , ns ,where us
(ξ(i))
are samples of eigenvectors obtained from eitherstochastic inverse subspace iteration or stochastic collocation.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 12 / 35
Numerical experiments: Vibration of undamped structures
We considered Young’s modulus
E (x , ξ) =M′∑`=0
E` (x)ψ` (ξ)
as a truncated lognormal process transformed from Gaussian r. process.Finite element spatial discretization leads to
K(ξ) u = λMu,
where K(ξ) =∑M′
`=0 K`ψ` (ξ) is the stochastic stiffness matrix,and M is the deterministic mass matrix. Using Cholesky M = LLT ,
L−1K(ξ) L−Tw = λw ,
A =M′∑`=0
A`ψ` (ξ) =M′∑`=0
[L−1K` (ξ) L
−T]ψ` (ξ) .
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 13 / 35
Numerical experiments: Timoshenko beam
Physical parameters: E0 = 108, ν = 0.30, with 20 finite elements (40 dof).
Stochastic parameters: N = 3, P = 3, Plog = 6, size of c`jk : 20× 20× 84,NMC = 5× 104, Smolyak sparse grid (Gauss-Hermite) with 69 points.
pdf of λ1 using only the stochastic Rayleigh quotient:
70 80 90 100 110 120 130 140 1500
0.01
0.02
0.03
0.04
0.05
0.06λ
Monte CarloStoch. CollocationStoch. Galerkin
0 50 100 150 200 2500
0.005
0.01
0.015
0.02
0.025λ
Monte CarloStoch. CollocationStoch. Galerkin
CoV = 10% CoV = 25%
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 14 / 35
Numerical experiments: Timoshenko beam (CoV = 25%)
The first ten coefficients of the gPC expansion of the smallest eigenvalueusing 0, 1 or 20 steps of stochastic inverse iteration, or stoch. collocation.Here d is the polynomial degree and k is the index of gPC basis function.
d k RQ(0) SII(1) SII(20) SC
0 0 103.0823 102.1705 102.1670 102.1713
11 14.0453 13.9402 13.9402 13.94082 -11.7568 -11.5862 -11.5859 -11.58483 5.1830 5.0654 5.0651 5.0669
2
4 1.4284 1.2919 1.2918 1.29095 -1.5368 -1.6766 -1.6767 -1.67646 0.5090 0.9030 0.9032 0.90357 1.1331 0.7533 0.7530 0.75298 -0.8696 -0.2965 -0.2960 -0.29559 0.5812 -0.2215 -0.2220 -0.2222
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 15 / 35
Numerical experiments: Timoshenko beam (CoV = 25%)
0 1 2 3 4 5 6x 104
0
1
2
3
4
5
6
7
x 10−4 λ
λ2
λ3
λ1 Monte CarloStoch. CollocationStoch. Galerkin
1 2 3 4 5 6 7x 105
0
1
2
3
4
5
6
7
8x 10−5 λ
λ3
λ4
λ5
Monte CarloStoch. CollocationStoch. Galerkin
pdf of λ1, λ2, λ3 pdf of λ3, λ4, λ5
0 5 10 15 20100
102
104
106
108
1010
iteration
ε0
Eigenpair:
12345
0 5 10 15 20104
106
108
1010
1012
1014
1016
1018
iteration
εσ 2
Eigenpair:
12345
convergence history of ε0 convergence history of εσ2
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 16 / 35
Numerical experiments: Timoshenko beam (CoV = 25%)
0 1 2 3 4 5 6 7x 105
0
0.5
1
1.5
2
x 10−5 λ
λ4
λ5
Monte CarloStoch. CollocationStoch. Galerkin
0 2 4 6 8 10x 10−4
0
2000
4000
6000
8000
10000
12000
εu
Stoch. CollocationStoch. Galerkin
pdf of λ4, λ5 pdf of εu for λ5
0 2 4 6 8 10x 10−10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 1010 εr
Stoch. CollocationStoch. Galerkin
0 2 4 6 8 10x 10−10
0
2
4
6
8
10
12
14
16x 109 εr
Stoch. CollocationStoch. Galerkin
pdf of εr for λ4 pdf of εr for λ5
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 17 / 35
Numerical experiments: Mindlin plate (CoV = 25%)
0 2 4 6 8 10 12 14x 104
0
0.5
1
1.5
2
2.5
3
3.5 x 10−4 λ
λ1
λ4
λ2,λ3
Monte CarloStoch. CollocationStoch. Galerkin
0 0.5 1 1.5 2x 10−3
0
2000
4000
6000
8000
10000
12000
εu
Stoch. CollocationStoch. Galerkin
pdf of λ1, λ2, λ3, λ4 pdf of εu for λ4
0 2 4 6 8 10x 10−3
0
50
100
150
200
250
300
350
400
450
500εr
Stoch. CollocationStoch. Galerkin
0 0.5 1 1.5 2x 10−5
0
1
2
3
4
5
6
7
8
9
x 105 εr
Stoch. CollocationStoch. Galerkin
pdf of εr for λ2 pdf of εr for λ4
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 18 / 35
II.) Numerical solution of Navier-Stokes equations with stochastic viscosity
The viscosity is given by a gPC expansion
ν ≡ ν(x , ξ) =Mν−1∑`=0
ν` (x)ψ` (ξ)
where {ν` (x)} is a set of given deterministic spatial functions.
Possible applications:1 the exact value of viscosity may not be known due to
measurement errorcontaminants with uncertain concentrationsmultiple phases with uncertain ratios
2 fluid properties might be influenced by an external field (MHD).
Equivalently, stochastic Reynolds number Re (ξ) = ULν(ξ) .
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 19 / 35
Stochastic Galerkin formulation
Idea: perform a Galerkin projection on the space TΓ (expectation E),and seek random fields for velocity ~u ∈ VE ⊗ TΓ, and pressure p ∈ QD ⊗ TΓ:
E[∫
D
ν∇~u : ∇~v +∫D
(~u · ∇~u) ~v −∫D
p (∇ · ~v)]
= E[∫
D
~f · ~v]∀~v ∈ VD ⊗ TΓ
E[∫
D
q (∇ · ~u)]
= 0 ∀q ∈ QD ⊗ TΓ
The stochastic counterpart of Newton iteration is
E[∫
D
ν∇δ~un : ∇~v + c(~un; δ~un, ~v) + c(δ~un; ~un, ~v)−∫D
δpn (∇ · ~v)]
= Rn
E[∫
D
q (∇ · δ~un)]
= rn
where
Rn (~v) = E[∫
D
~f · ~v −∫D
ν∇~un : ∇~v − c(~un; ~un, ~v) +∫D
pn (∇ · ~v)]
rn (q) = −E[∫
D
q (∇ · ~un)].
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 20 / 35
Stochastic Galerkin FEM: ordering of unknowns
We seek a discrete approximation of the solution of the form
~u (x , ξ) ≈M−1∑k=0
Nu∑i=1
uikφi (x)ψk(ξ) =M−1∑k=0
~uk(x)ψk(ξ)
p (x , ξ) ≈M−1∑k=0
Np∑j=1
pjkϕj(x)ψk(ξ) =M−1∑k=0
~pk(x)ψk(ξ)
Structure of operators depends on the ordering of the coeffs {uik}, {pjk}.We will group velocity-pressure pairs for each k , the index of stochasticbasis functions, giving the ordered list of coefficients
u1:Nu ,0, p1:Np ,0, u1:Nu ,1, p1:Np ,1, . . . , u1:Nu ,M−1, p1:Np ,M−1.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 21 / 35
Stochastic Galerkin FEM: structure of Stokes operator
The discrete stochastic Stokes operator is built using matrices
A`= [a`,ab] , a`,ab =
(∫D
ν` (x) ∇φb : ∇φa), ` = 1, . . . ,Mν − 1
which are incorporated into the block matrices
S0 =[
A0 BT
B 0
], S` =
[A` 00 0
], ` = 1, . . . ,Mν − 1.
These operators will be coupled with matrices arising from terms in TP ,
H` = [h`,jk ] , h`,jk ≡ E [ψ`ψjψk ] , ` = 0, . . . ,Mν − 1, j , k = 0, . . . ,M − 1.
The discrete stochastic (Galerkin) Stokes system is(Mν−1∑`=0
H` ⊗ S`
)v = y
where ⊗ corresponds to the matrix Kronecker product.Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 22 / 35
Stochastic Galerkin FEM: block saddle-point structure
The stochastic (Galerkin) Stokes matrix(Mν−1∑`=0
H` ⊗ S`
)
contains a set of M block 2× 2 matrices of saddle-point structure
S0 +Mν−1∑`=1
h`,jjS`, j = 0, . . . ,M − 1.
along its block diagonal =⇒ enables use of existing deterministicsolvers for the individual diagonal blocks.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 23 / 35
Stochastic Galerkin FEM: linearized N-S operators
Step n of the stochastic nonlinear iteration entailsM̂−1∑`=0
H` ⊗Fn`
δvn = Rn vn+1 = vn + δvnwhere
Rn = y −
M̂−1∑`=0
H` ⊗ Pn`
vnand vn and δvn are current velocity and pressure coefficients and updates.Here
Fn0 =[
Fn0 BT
B 0
]Fn` =
[F` 00 0
]
Fn` = A` + Nn` + W`
n for the stochastic Newton method,
Fn` = A` + Nn` for stochastic Picard iteration.
Question: How are the matrices A`, Nn` , W`n computed?
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 24 / 35
Stochastic Galerkin FEM: components of N-S operators
At step n of the nonlinear iteration, let
~un` (x) the `th term of the velocity iterate, ` = 1, . . . ,M
and let
Nn` =[nn`,ab
]nn`,ab =
∫D
(~un` · ∇φb) · φa
Wn` =[wn`,ab
]wn`,ab =
∫D
(φb · ∇~un` ) · φa
and as before
A`= [a`,ab] , a`,ab =
(∫Dν` (x) ∇φb : ∇φa
)` = 1, . . . ,Mν − 1
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 25 / 35
Model problem: Flow around an obstacle
Flow around an obstacle with 12 640 velocity and 1640 pressure dofs.
0 2 4 6 8 10 12−1
−0.5
0
0.5
1
Stochastic Galerkin method implemented using the IFISS 3.3 package.1
Viscosity with mean ν0 = 1/50 or 1/150 and lognormal distribution,
CoV =σνν0, Re0 =
U L
ν0= 100 or 300.
1http://www.cs.umd.edu/~elman/ifiss.htmlSoused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 26 / 35
http://www.cs.umd.edu/~elman/ifiss.html
Model problem: Random viscosity
Viscosity ν: a truncated lognormal process (ν0 = 1/50 or 1/150), itsrepresentation computed from Gaussian random process (Ghanem, 1999)with the covariance function
C (X1,X2) = σ2g exp
(−|x2 − x1|
Lx− |y2 − y1|
Ly
)Lx and Ly : correlation lengths of ξi in the x and y directions, respectively,
(set to be equal to 25% of the domain size, i.e. Lx = 3 and Ly = 0.5σg : the standard deviation of the Gaussian random field.
N: the stochastic dimension (N = 2)P: the degree for the polynomial expansion of the solution (P = 3),
the degree for the expansion of the lognormal process was 2PCoV = σν/ν0: coefficient of variation of the lognormal field (10% or 30%).
=⇒ M = 10, Mν = 28 and 142, 800 dof in the global stochastic matrix.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 27 / 35
Model problem: Nonlinear solvers
Hybrid scheme: Stokes → Picard → Newton(direct solvers for the linear systems, rel. res. 10−8)
Re = 100, CoV = 10%: 6 Picard steps 1 Newton step(s)Re = 100, CoV = 30%: 6 3Re = 300, CoV = 10%: 20 1Re = 300, CoV = 30%: 20 2
Variant with an inexact Picard iteration (block diagonal matrix H0 ⊗Fn0 ):for CoV = 10% at most one extra Newton step (both Re = 100 and 300),for CoV = 30% this inexact method failed to converge.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 28 / 35
Model problem: Mean velocity and pressure (Re = 100 and CoV = 10%)
Mean of horizontal and vertical velocity:
0 2 4 6 8 10 12−1
0
1
0
0.5
1
0 2 4 6 8 10 12−1
0
1
−0.5
0
0.5
Mean of pressure:
0 2 4 6 8 10 12−1
0
10
0.5
1
1.5
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 29 / 35
Model problem: Variance of velocity and pressure (Re = 100, CoV = 10%)
Variance of horizontal and vertical velocity:
0 2 4 6 8 10 12−1
0
1
1234
x 10−3
0 2 4 6 8 10 12−1
0
1
5
10
15x 10−4
Variance of pressure:
0 2 4 6 8 10 12−1
0
1
5
10
15
x 10−3
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 30 / 35
Model problem: Mean velocity and pressure (Re = 300 and CoV = 10%)
Mean of horizontal and vertical velocity:
0 2 4 6 8 10 12−1
0
1
0
0.5
1
0 2 4 6 8 10 12−1
0
1
−0.5
0
0.5
Mean of pressure:
0 2 4 6 8 10 12−1
0
1−0.5
0
0.5
1
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 31 / 35
Model problem: Variance of velocity and pressure (Re = 300, CoV = 10%)
Variance of horizontal and vertical velocity:
0 2 4 6 8 10 12−1
0
1
1
2
3
x 10−3
0 2 4 6 8 10 12−1
0
1
2468101214
x 10−4
Variance of pressure:
0 2 4 6 8 10 12−1
0
10.5
1
1.5
2
x 10−3
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 32 / 35
Model problem: Variance of velocity and pressure (Re = 300, CoV = 30%)
Variance of horizontal and vertical velocity:
0 2 4 6 8 10 12−1
0
1
0.010.020.030.04
0 2 4 6 8 10 12−1
0
1
24681012
x 10−3
Variance of pressure:
0 2 4 6 8 10 12−1
0
15
10
15
x 10−3
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 33 / 35
Model problem: pdf of horizontal velocity and pressure at point (3.6436, 0)
CoV = 10% CoV = 30%
ux
−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50
1
2
3
4
5
6
7
MCSCSG
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80
0.5
1
1.5
2
2.5
MCSCSG
p
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.250
2
4
6
8
10
12
MCSCSG
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60
0.5
1
1.5
2
2.5
3
3.5
4
MCSCSG
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 34 / 35
References
Beďrich Soused́ık, Howard C. Elman:Inverse subspace iteration for spectral stochastic finite element methods,SIAM/ASA Journal on Uncertainty Quantification 4(1), 163–189, 2016.
Beďrich Soused́ık, Howard C. Elman:Stochastic Galerkin methods for the steady-state Navier-Stokes equations,Journal of Computational Physics 316, 435–452, 2016.
Soused́ık & Elman (U. of Maryland) SSFEM for two NL problems SISSA, July 2017 35 / 35
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