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
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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