Kernel-based Collocation Methods versus Galerkin Finite Element
STOCHASTIC GALERKIN FINITE ELEMENT METHODS FOR...
Transcript of STOCHASTIC GALERKIN FINITE ELEMENT METHODS FOR...
STOCHASTIC GALERKIN FINITE ELEMENT METHODS FOR
SADDLE POINT PROBLEMS WITH RANDOM DATA
Alex Bespalov, Catherine Powell, David Silvester
School of Mathematics, University of Manchester,
Manchester, United Kingdom
Workshop “Numerical Analysis of Stochastic PDEs”
Mathematics Institute, University of Warwick
11 – 12 June, 2012
A. Bespalov ∗ sGFEM for saddle point problems with random data 1/22
What is this talk about...
∗ Saddle point problems with random data
∗ Stochastic Galerkin mixed finite element method
∗ Inf-sup stability of discrete problem, solution regularity, error analysis
A. Bespalov ∗ sGFEM for saddle point problems with random data 2/22
Saddle point problems
Find (u, p) ∈ V ×W such that
a(u, v) + b(v, p) = f(v) ∀ v ∈ V,
b(u, q) = g(q) ∀ q ∈ W.
Here, V and W represent Hilbert spaces;
a : V × V → IR is a symmetric bounded bilinear form,
b : V ×W → IR is a bounded bilinear form and
f : V → IR and g : W → IR are linear functionals.
A. Bespalov ∗ sGFEM for saddle point problems with random data 3/22
Saddle point problems with random data
Random coefficient(s): find (u, p) ∈ V ×W such that
a(u, v) + b(v, p) = f(v) ∀ v ∈ V,
b(u, q) = g(q) ∀ q ∈ W.
Examples: groundwater flow modelling, steady state Navier-Stokes flow
A. Bespalov ∗ sGFEM for saddle point problems with random data 4/22
Saddle point problems with random data
Random coefficient(s): find (u, p) ∈ V ×W such that
a(u, v) + b(v, p) = f(v) ∀ v ∈ V,
b(u, q) = g(q) ∀ q ∈ W.
Random domain: find (u, p) ∈ V ×W such that
a(u, v) + b(v, p) = f(v) ∀ v ∈ V,
b(u, q) = g(q) ∀ q ∈ W.
Fictitious domain approach for elliptic PDEs in random domains:
[Canuto and Kozubek ’07].
A. Bespalov ∗ sGFEM for saddle point problems with random data 4/22
Saddle point problems with random data
Random coefficient(s): find (u, p) ∈ V ×W such that
a(u, v) + b(v, p) = f(v) ∀ v ∈ V,
b(u, q) = g(q) ∀ q ∈ W.
Random domain: find (u, p) ∈ V ×W such that
a(u, v) + b(v, p) = f(v) ∀ v ∈ V,
b(u, q) = g(q) ∀ q ∈ W.
Random forces and/or boundary conditions: find (u, p) ∈ V ×W such that
a(u, v) + b(v, p) = f(v) ∀ v ∈ V,
b(u, q) = g(q) ∀ q ∈ W.
A. Bespalov ∗ sGFEM for saddle point problems with random data 4/22
Example: steady flow over a step with data uncertainty
Model problem:
−ν∇2u + u · ∇u +∇p = f in D,
∇ · u = 0 in D,
u = g on ∂DDir,
ν∇u · n− n p = 0 on ∂DNeu.
Figure 1. The backward-facing step domain.
A. Bespalov ∗ sGFEM for saddle point problems with random data 5/22
Example: steady flow over a step with data uncertainty
Model problem:
−ν∇2u + u · ∇u +∇p = f in D,
∇ · u = 0 in D,
u = g on ∂DDir,
ν∇u · n− n p = 0 on ∂DNeu.
We can model uncertainty in the viscosity as ν(ω) = ν0 + ν1ξ1(ω).
If ξ1 ∼ U(−√3,√
3), then ν is a uniform random variable with
E[ν(ω)] = ν0, Var[ν(ω)] = ν21 .
A. Bespalov ∗ sGFEM for saddle point problems with random data 5/22
Example: steady flow over a step with data uncertainty
Model problem:
−ν∇2u + u · ∇u +∇p = f in D,
∇ · u = 0 in D,
u = g on ∂DDir,
ν∇u · n− n p = 0 on ∂DNeu.
We can model uncertainty in the viscosity as ν(ω) = ν0 + ν1ξ1(ω).
If ξ1 ∼ U(−√3,√
3), then ν is a uniform random variable with
E[ν(ω)] = ν0, Var[ν(ω)] = ν21 .
Then ν ∼ U(νmin, νmax) with νmin = ν0 − ν1
√3, νmax = ν0 + ν1
√3, and
Re(ω) =constν(ω)
, E[Re] = const E[ν−1] =const∗
ν1log
(νmax
νmin
).
A. Bespalov ∗ sGFEM for saddle point problems with random data 5/22
Example: steady flow over a step with data uncertainty
Random viscosity: ν(ω) = ν0 + ν1ξ1(ω) with ν0 = 1/50 and ν1 = 1/500.
Figure 2. Streamlines of the mean flow field (top) and contours of the variance of
the magnitude of flow field (bottom).
A. Bespalov ∗ sGFEM for saddle point problems with random data 6/22
Example: steady flow over a step with data uncertainty
Random viscosity: ν(ω) = ν0 + ν1ξ1(ω) with ν0 = 1/50 and ν1 = 1/500.
mean pressure field
variance of the pressure field
Figure 3. The mean (top) and the variance (bottom) of the pressure field.
A. Bespalov ∗ sGFEM for saddle point problems with random data 7/22
Example: steady flow over a step with data uncertainty
More details on this problem (including stochastic Galerkin mixed finite element
scheme, properties of saddle point linear systems, and analysis of precondition-
ing strategies):
D. Silvester, A. B. and C. Powell, A framework for the development of implicit
solvers for incompressible flow problems, Discrete and Continuous Dynamical
Systems - Series S, 2012 (to appear).
C. Powell and D. Silvester, Preconditioning steady-state Navier-Stokes equa-
tions with random data, MIMS EPrint 2012.35, The University of Manchester,
2012 (submitted).
A. Bespalov ∗ sGFEM for saddle point problems with random data 8/22
Model problem
D ⊂ Rd (d = 2, 3) – spatial domain;
(Ω,F ,P) – complete probability space;
A−1(x, ω) : D × Ω → R – second-order correlated random field.
Model problem:
find random fields p(x, ω) and u(x, ω) such that P-almost everywhere in Ω
A−1 (x, ω)u (x, ω)−∇p (x, ω) = 0 x ∈ D,
∇ · u (x, ω) = 0 x ∈ D,
p (x, ω) = g(x) x ∈ ∂DDir,
u (x, ω) · n = 0 x ∈ ∂DNeu.
A. Bespalov ∗ sGFEM for saddle point problems with random data 9/22
Useful references
• Primal formulations, stochastic Galerkin FEM, error analysis
[Babuska, Tempone and Zouraris ’04],
[Frauenfelder, Schwab, Todor ’05].
• Stochastic collocation FEM, mixed formulation,
log-normal distribution of random data
[Ganis, Klie, Wheeler, Wildey, Yotov, and Zhang ’08].
• Stochastic Galerkin FEM, mixed formulation,
linear algebra and fast solvers
[Ernst, Powell, Silvester, and Ullmann ’09],
[Elman, Furnival, and Powell ’10].
A. Bespalov ∗ sGFEM for saddle point problems with random data 10/22
Weak formulation
X(D) – a Banach space of real-valued functions on D with norm ‖ · ‖X(D).
Vector spaces of random fields
L2P(Ω, X(D)) :=
v (x, ω) ; v : D × Ω → R,
‖v‖L2P(Ω,X(D)) :=
(E
[‖v‖2X(D)
])1/2
< ∞
;
V := L2P(Ω,H0(div, D)) and W := L2
P(Ω, L2(D)).
A. Bespalov ∗ sGFEM for saddle point problems with random data 11/22
Weak formulation
X(D) – a Banach space of real-valued functions on D with norm ‖ · ‖X(D).
Vector spaces of random fields
L2P(Ω, X(D)) :=
v (x, ω) ; v : D × Ω → R,
‖v‖L2P(Ω,X(D)) :=
(E
[‖v‖2X(D)
])1/2
< ∞
;
V := L2P(Ω,H0(div, D)) and W := L2
P(Ω, L2(D)).
Weak formulation:
find u(x, ω) ∈ V and p(x, ω) ∈ W such that
E[(
A−1(x, ω)u,v)]
+ E [(p,∇ · v)] = E[(g,v · n)∂DDir
],
E [(w,∇ · u)] = 0
for all v(x, ω) ∈ V and w(x, ω) ∈ W.
A. Bespalov ∗ sGFEM for saddle point problems with random data 11/22
Discretisation strategy
Discretisation method: stochastic Galerkin mixed finite elements.
Three levels of approximation
• Approximation of random data, A−1 (x, ω) ≈ A−1M (x, ξ(ω)):
e.g., using the truncated Karhunen-Loeve expansion of A−1 (x, ω);
• Spatial discretisation on D:
e.g., by the lowest-order mixed FEM with mesh-size h;
• Discretisation on Γ = ξ(Ω) ⊂ RM :
e.g., global polynomial approximation of total degree ≤ k.
A. Bespalov ∗ sGFEM for saddle point problems with random data 12/22
Discretisation strategy
Discretisation method: stochastic Galerkin mixed finite elements.
Three levels of approximation
• Approximation of random data, A−1 (x, ω) ≈ A−1M (x, ξ(ω)):
e.g., using the truncated Karhunen-Loeve expansion of A−1 (x, ω);
• Spatial discretisation on D:
e.g., by the lowest-order mixed FEM with mesh-size h;
• Discretisation on Γ = ξ(Ω) ⊂ RM :
e.g., global polynomial approximation of total degree ≤ k.
Three levels of approximation =⇒ three discretisation parameters (M, h, k)
and three sources of error.
A. Bespalov ∗ sGFEM for saddle point problems with random data 12/22
Approximation of random data
A−1(x, ω) ≈ A−1M (x, ω)
... leads to
Perturbed weak formulation:
find uM (x, ω) ∈ V and pM (x, ω) ∈ W such that
E[(
A−1M (x, ω)uM ,v
)]+ E [(pM ,∇ · v)] = E
[(g,v · n)∂DDir
],
E [(w,∇ · uM )] = 0
for all v(x, ω) ∈ V and w(x, ω) ∈ W.
A. Bespalov ∗ sGFEM for saddle point problems with random data 13/22
Estimating the perturbation error
Lemma 1. Assume that
0 < Amin ≤ A−1(x, ω) ≤ Amax < ∞ a. e. in D × Ω,
0 < AMmin ≤ A−1
M (x, ω) ≤ AMmax < ∞ a. e. in D × Ω.
Then there exist unique solution pairs (u, p) ∈ V×W, (uM , pM ) ∈ V×W and
‖u− uM‖V + ‖p− pM‖W ≤ C ‖A−1 −A−1M ‖L∞(D×Ω). (1)
A. Bespalov ∗ sGFEM for saddle point problems with random data 14/22
Estimating the perturbation error
Lemma 1. Assume that
0 < Amin ≤ A−1(x, ω) ≤ Amax < ∞ a. e. in D × Ω,
0 < AMmin ≤ A−1
M (x, ω) ≤ AMmax < ∞ a. e. in D × Ω.
Then there exist unique solution pairs (u, p) ∈ V×W, (uM , pM ) ∈ V×W and
‖u− uM‖V + ‖p− pM‖W ≤ C ‖A−1 −A−1M ‖L∞(D×Ω). (1)
Remark 1. The constant C in (1) depends on Amin, Amax, AMmin, AM
max, on
the inf-sup constant and the Dirichlet data...
A. Bespalov ∗ sGFEM for saddle point problems with random data 14/22
Estimating the perturbation error
Lemma 1. Assume that
0 < Amin ≤ A−1(x, ω) ≤ Amax < ∞ a. e. in D × Ω,
0 < AMmin ≤ A−1
M (x, ω) ≤ AMmax < ∞ a. e. in D × Ω.
Then there exist unique solution pairs (u, p) ∈ V×W, (uM , pM ) ∈ V×W and
‖u− uM‖V + ‖p− pM‖W ≤ C ‖A−1 −A−1M ‖L∞(D×Ω). (1)
Remark 1. The constant C in (1) depends on Amin, Amax, AMmin, AM
max, on
the inf-sup constant and the Dirichlet data...
...but if ‖A−1 − A−1M ‖L∞(D×Ω) → 0 as M →∞ (see next Lemma), then,
for sufficiently large M , we can set
AMmin := 1
2Amin, AMmax := Amax + 1
2Amin.
Then, the constant C in (1) is independent of M .
A. Bespalov ∗ sGFEM for saddle point problems with random data 14/22
Estimating the error in approximation of random data
Goal: upper bound for ‖A−1 −A−1M ‖L∞(D×Ω).
A. Bespalov ∗ sGFEM for saddle point problems with random data 15/22
Estimating the error in approximation of random data
Goal: upper bound for ‖A−1 −A−1M ‖L∞(D×Ω).
Representation of random data using Karhunen-Loeve (KL) expansion:
A−1(x, ω) = E[A−1](x) +∑∞
n=1
√λn ϕn(x) ξn(ω)
≈ E[A−1](x) +∑M
n=1
√λnϕn(x) ξn(ω) =: A−1
M (x, ω).
Lemma 2 [Frauenfelder, Schwab, Todor ’05]. Assume:
(i) ξn∞n=1 is uniformly bounded;
(ii) covariance function C[A−1](x,x′) is (piecewise) analytic on D×D. Then
‖A−1 −A−1M ‖L∞(D×Ω) ≤ Ce−cM1/d
.
A. Bespalov ∗ sGFEM for saddle point problems with random data 15/22
Estimating the error in approximation of random data
Goal: upper bound for ‖A−1 −A−1M ‖L∞(D×Ω).
Representation of random data using Karhunen-Loeve (KL) expansion:
A−1(x, ω) = E[A−1](x) +∑∞
n=1
√λn ϕn(x) ξn(ω)
≈ E[A−1](x) +∑M
n=1
√λnϕn(x) ξn(ω) =: A−1
M (x, ω).
Lemma 2 [Frauenfelder, Schwab, Todor ’05]. Assume:
(i) ξn∞n=1 is uniformly bounded;
(ii) covariance function C[A−1](x,x′) is (piecewise) analytic on D×D. Then
‖A−1 −A−1M ‖L∞(D×Ω) ≤ Ce−cM1/d
.
Lemma 1 + Lemma 2 =⇒Theorem 1. ‖u− uM‖V + ‖p− pM‖W = O
(e−cM1/d
).
A. Bespalov ∗ sGFEM for saddle point problems with random data 15/22
More assumptions ...
We further assume that
• random variables ξn : Ω → R (n = 1, 2, . . .) are independent;
• images Γn = ξn(Ω) are bounded intervals in R;
• ∃ ρn : Γn → R+ – a density function of ξn (n = 1, . . . , M);
A. Bespalov ∗ sGFEM for saddle point problems with random data 16/22
More assumptions ...
We further assume that
• random variables ξn : Ω → R (n = 1, 2, . . .) are independent;
• images Γn = ξn(Ω) are bounded intervals in R;
• ∃ ρn : Γn → R+ – a density function of ξn (n = 1, . . . , M);
... then
• uM (x, ω) = uM (x, ξ1(ω), . . . , ξM (ω)),pM (x, ω) = pM (x, ξ1(ω), . . . , ξM (ω));
A. Bespalov ∗ sGFEM for saddle point problems with random data 16/22
More assumptions ...
We further assume that
• random variables ξn : Ω → R (n = 1, 2, . . .) are independent;
• images Γn = ξn(Ω) are bounded intervals in R;
• ∃ ρn : Γn → R+ – a density function of ξn (n = 1, . . . , M);
... then
• uM (x, ω) = uM (x, ξ1(ω), . . . , ξM (ω)),pM (x, ω) = pM (x, ξ1(ω), . . . , ξM (ω));
• ρ(y) :=∏M
n=1 ρn – the joint probability density of (ξ1, . . . , ξM ), where
y = (y1, . . . ,yM ) ∈ Γ with yn = ξn(ω) (n = 1, . . . ,M), and
Γ = supp ρ = Γ1 × . . .× ΓM ⊂ RM ;
A. Bespalov ∗ sGFEM for saddle point problems with random data 16/22
More assumptions ...
We further assume that
• random variables ξn : Ω → R (n = 1, 2, . . .) are independent;
• images Γn = ξn(Ω) are bounded intervals in R;
• ∃ ρn : Γn → R+ – a density function of ξn (n = 1, . . . , M);
... then
• uM (x, ω) = uM (x, ξ1(ω), . . . , ξM (ω)),pM (x, ω) = pM (x, ξ1(ω), . . . , ξM (ω));
• ρ(y) :=∏M
n=1 ρn – the joint probability density of (ξ1, . . . , ξM ), where
y = (y1, . . . ,yM ) ∈ Γ with yn = ξn(ω) (n = 1, . . . ,M), and
Γ = supp ρ = Γ1 × . . .× ΓM ⊂ RM ;
• (Ω,F ,P) can be replaced by(Γ,B(Γ), ρ(y)dy
);
• for any measurable ϕ = ϕ(ξ1, . . . , ξM ), one has E[ϕ] =∫Γ
ϕ(y) ρ(y)dy.
A. Bespalov ∗ sGFEM for saddle point problems with random data 16/22
... and another weak formulation
Denote
V := L2ρ(Γ,H0(div; D)), W := L2
ρ(Γ, L2(D));
aM (u,v) =(A−1
M u,v), b(p,v) = (p,∇ · v); `(v) = (g,v · n)∂DDir
.
Parametric deterministic formulation:
find uM (x,y) ∈ V and pM (x,y) ∈ W such that
E [aM (uM ,v)] + E [b (pM ,v)] = E [`(v)] ,
E [b (w,uM )] = 0
for all v(x,y) ∈ V and w(x,y) ∈ W.
Remark 2. This problem is uniquely solvable under the assumptions in the
statement of Lemma 1.
A. Bespalov ∗ sGFEM for saddle point problems with random data 17/22
Stochastic Galerkin mixed FEM
Discrete subspaces
(i) on the spatial domain D ⊂ Rd: Xdivh ⊂ H0(div; D), X0
h ⊂ L2(D);(ii) on the outcomes set Γ ⊂ RM : Sk ⊂ L2
ρ(Γ).
A. Bespalov ∗ sGFEM for saddle point problems with random data 18/22
Stochastic Galerkin mixed FEM
Discrete subspaces
(i) on the spatial domain D ⊂ Rd: Xdivh ⊂ H0(div; D), X0
h ⊂ L2(D);(ii) on the outcomes set Γ ⊂ RM : Sk ⊂ L2
ρ(Γ).
Discrete formulation (sGFEM):
find uhk(x,y) ∈ Xdivh ⊗ Sk and phk(x,y) ∈ X0
h ⊗ Sk satisfying
E [aM (uhk,v)] + E [b (phk,v)] = E [` (v)] ,
E [b (w,uhk)] = 0
for all v ∈ Xdivh ⊗ Sk and w ∈ X0
h ⊗ Sk.
A. Bespalov ∗ sGFEM for saddle point problems with random data 18/22
Stochastic Galerkin mixed FEM
Discrete subspaces
(i) on the spatial domain D ⊂ Rd: Xdivh ⊂ H0(div; D), X0
h ⊂ L2(D);(ii) on the outcomes set Γ ⊂ RM : Sk ⊂ L2
ρ(Γ).
Discrete formulation (sGFEM):
find uhk(x,y) ∈ Xdivh ⊗ Sk and phk(x,y) ∈ X0
h ⊗ Sk satisfying
E [aM (uhk,v)] + E [b (phk,v)] = E [` (v)] ,
E [b (w,uhk)] = 0
for all v ∈ Xdivh ⊗ Sk and w ∈ X0
h ⊗ Sk.
Theorem 2. Let Xdivh , X0
h be a deterministic inf-sup stable pairing with
discrete inf-sup constant β. Then, for any choice of Sk ⊂ L2ρ(Γ), the pairing(
Xdivh ⊗ Sk
),(X0
h ⊗ Sk
)for the sGFEM is inf-sup stable with the same discrete
inf-sup constant β, and the sGFEM converges quasi-optimally.
A. Bespalov ∗ sGFEM for saddle point problems with random data 18/22
Estimating the stochastic Galerkin error
Total error of the stochastic Galerkin FEM:
Ehk := ‖uM − uhk‖V + ‖pM − phk‖W .
A. Bespalov ∗ sGFEM for saddle point problems with random data 19/22
Estimating the stochastic Galerkin error
Total error of the stochastic Galerkin FEM:
Ehk := ‖uM − uhk‖V + ‖pM − phk‖W .
Decomposition of the error
Ehk ¹ infv∈Xdiv
h ⊗Sk
‖uM − v‖V + infw∈X0
h⊗Sk
‖pM − w‖W
¹ ‖uM −Πdivh uM‖V + ‖pM −Π0
h pM‖W
+ ‖uM −Π0,ρk uM‖V + ‖pM −Π0,ρ
k pM‖W
=‘spatial’ h-error
+
‘stochastic’ k-error
,
Πdivh is an H(div; D)-conforming interpolation operator (defined elementwise),
Π0h is L2(D)-projector onto X0
h ⊂ L2(D),
Π0,ρk is L2
ρ(Γ)-orthogonal projection onto Sk ⊂ L2ρ(Γ).
A. Bespalov ∗ sGFEM for saddle point problems with random data 19/22
Regularity of the solution
Parameterised coefficient:
A−1M (x,y) = E[A−1](x) +
∑Mn=1
√λn ϕn(x) yn.
Spatial regularity
If E[A−1] ∈ C1(D) and C[A−1] is smooth on D ×D, then ∃ r > 0 such that(uM , pM
) ∈ L2ρ(Γ;Hr(div, D))× L2
ρ(Γ; Hr(D)).
A. Bespalov ∗ sGFEM for saddle point problems with random data 20/22
Regularity of the solution
Parameterised coefficient:
A−1M (x,y) = E[A−1](x) +
∑Mn=1
√λn ϕn(x) yn.
Spatial regularity
If E[A−1] ∈ C1(D) and C[A−1] is smooth on D ×D, then ∃ r > 0 such that(uM , pM
) ∈ L2ρ(Γ;Hr(div, D))× L2
ρ(Γ; Hr(D)).
Regularity with respect to y1, . . . , yM
Lemma 4. If Γn (n ∈ 1, 2, . . . ,M) is a bounded interval in R then the
functions uM and pM , as functions of variable yn, can be analytically extended
to the same region of the complex plane:
A. Bespalov ∗ sGFEM for saddle point problems with random data 20/22
Regularity of the solution
Parameterised coefficient:
A−1M (x,y) = E[A−1](x) +
∑Mn=1
√λn ϕn(x) yn.
Spatial regularity
If E[A−1] ∈ C1(D) and C[A−1] is smooth on D ×D, then ∃ r > 0 such that(uM , pM
) ∈ L2ρ(Γ;Hr(div, D))× L2
ρ(Γ; Hr(D)).
Regularity with respect to y1, . . . , yM
Lemma 4. If Γn (n ∈ 1, 2, . . . ,M) is a bounded interval in R then the
functions uM and pM , as functions of variable yn, can be analytically extended
to the same region of the complex plane:
Σn :=
z ∈ C; |z − y0
n| <infx∈D
(A−1
M (x, y0n,y∗n)
)√
λn ‖ϕn‖L∞(D)
∀ y0n ∈ Γj
,
where y∗n = (y1, . . . , yn−1, yn+1, . . . , yM ) for any n ∈ 1, 2, . . . , M.
A. Bespalov ∗ sGFEM for saddle point problems with random data 20/22
Estimating the stochastic Galerkin error
Theorem 3. Assume:
(i) KL-expansion of A−1 with uniformly distributed random variables ξn;
(ii) given k = (k1, . . . , kM ) ∈ NM0 , Sk := Sk1(Γ1)⊗ . . .⊗ SkM (ΓM );
(iii) technical coercivity assumption. Then there holds
‖uM − uhk‖V + ‖pM − phk‖W ≤ C
(hmin r,1 +
M∑n=1
ηnkn+1
),
where ηn =(χn +
√χ2
n − 1)−1
∈ (0, 1) with χn = 1 + const√λn ‖ϕn‖L∞(D)
for n = 1, . . . ,M .
A. Bespalov ∗ sGFEM for saddle point problems with random data 21/22
References
More details of this work:
A. B., C. Powell and D. Silvester, A priori error analysis of stochastic Galerkin
mixed approximations of elliptic PDEs with random data, SIAM J. Numer.
Anal., 2012 (to appear).
Useful references
[1] I. Babuska, R. Tempone and G. E. Zouraris, SINUM, 42 (2004).
[2] H.C. Elman, D.G. Furnival and C.E. Powell, Math. Comp., 79 (2010).
[3] O.G. Ernst, C.E. Powell, D.J. Silvester and E. Ullmann, SISC, 31 (2009).
[4] P. Frauenfelder, C. Schwab and R.A.Todor, CMAME, 194 (2005).
[5] B. Ganis, H. Klie, M. Wheeler, T. Wildey, I. Yotov and D. Zhang, CMAME,
197 (2008).
A. Bespalov ∗ sGFEM for saddle point problems with random data 22/22
References
More details of this work:
A. B., C. Powell and D. Silvester, A priori error analysis of stochastic Galerkin
mixed approximations of elliptic PDEs with random data, SIAM J. Numer.
Anal., 2012 (to appear).
Useful references
[1] I. Babuska, R. Tempone and G. E. Zouraris, SINUM, 42 (2004).
[2] H.C. Elman, D.G. Furnival and C.E. Powell, Math. Comp., 79 (2010).
[3] O.G. Ernst, C.E. Powell, D.J. Silvester and E. Ullmann, SISC, 31 (2009).
[4] P. Frauenfelder, C. Schwab and R.A.Todor, CMAME, 194 (2005).
[5] B. Ganis, H. Klie, M. Wheeler, T. Wildey, I. Yotov and D. Zhang, CMAME,
197 (2008).
Thank you for your attention!
A. Bespalov ∗ sGFEM for saddle point problems with random data 22/22