Stochastic Galerkin Methods without Uniform Ellipticity Marcus Sarkis (WPI/IMPA) Collaborator: Juan Galvis (Texas A & M) WPI/IMPA RICAM MS & AEE-Workshop4, Dec13/2011 Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis SPDE-GALERKIN RICAM 2011 1 / 28
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Stochastic Galerkin Methods without Uniform Ellipticity€¦ · without uniform ellipticity,Juan Galvisand Marcus Sarkis. SIAM J. Numer. Anal., Vol. 47(5), pp. 3624-3651, 2009. Regularity
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Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 1 / 28
Problem of interest
Consider the Darcy’s equation−∇x . (κ(x , ω)∇xu(x , ω)) = f (x , ω), for x ∈ D ⊂ Rd
u(x , ω) = 0, on ∂D
κ(x , ω) = eW (x ,ω)
I W (x , ω) =∑∞
k=1 ak(x)ξk(ω)
I ξk are iid standard normal random variables
I eW (x,ω) ∈ (0,∞) not bounded, not uniformly elliptic
f (x , ω)
I Random forcing term
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 2 / 28
Outline
One-dimensional log-normal noise
White noise framework
Countable infinite-dimensional log-normal noise
Galerkin spectral method
Discretization, well-posedness, a priori error
Numerical results
Conclusions
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 3 / 28
Breeding analysis on Log-Normal without ellipticity
November 2005: I. Babuska, F. Nobile and R. Tempone: A stochasticcollocation method for elliptic partial differential equations withrandom input data.
March 2008: J. Galvis and S., Approximating infinity-dimensionalstochastic Darcy’s equations without uniform ellipticity.
March 2009: X. Wan, B. Rozovskii and G. E. Karniadakis, Astochastic modeling methodology based on weighted Wiener chaosand Malliavin calculus.
May 2009: C.J. Gittelson, Stochastic Galerkin discretization of thelognormal isotropic diffusion problem.
June 2010: J. Charrier, Strong and weak error estimates for thesolutions of elliptic partial differential equations with randomcoefficients.
January 2011: A. Mugler and H.-J. Starkloff, On elliptic partialdifferential equations with random coefficients.
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 4 / 28
References
Approximating infinity-dimensional stochastic Darcy’s equationswithout uniform ellipticity, Juan Galvis and Marcus Sarkis. SIAM J.Numer. Anal., Vol. 47(5), pp. 3624-3651, 2009.
Regularity results for the ordinary product stochastic pressureequation, Juan Galvis and Marcus Sarkis. Submitted. Preprint serieIMPA A 692, 2011.
An introduction to infinite-dimensional analysis, Giuseppe Da Prato.Universitext, Springer-Verlag, Berlin, 2006.
Errors for K = N = k, h = 1/16, 1/32 and ε = 12 , s = 0. For
h = 1/32 we have added in parenthesis the reduction factor, whenpassing to next value of k, corresponding to the projection and finiteelement error in the seminorm | · |U1
0and the finite element error in
the κ-energy norm.
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 26 / 28
Numerical experiments
Approximation of u(0,0,0,... ) for K = N = 3, h = 110 and ε = 0, 1, 2
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 27 / 28
Conclusions
Ellipticity treatment
Unified framework for KL and smoothed white noise
More general f and infinite-dimensional case
Weighted norms, well-posedness, a priori error estimates
Framework for establishing regularity theory
Marcus Sarkis (WPI/IMPA), Collaborator: Juan Galvis (Texas A & M) ()SPDE-GALERKIN RICAM 2011 28 / 28