cao's numerical method for solving SPDE

Topics: Numerical analysis, Finite element method, Partial differential equation Pages: 35 (1985 words) Published: October 11, 2013
FINITE ELEMENT APPROXIMATIONS FOR
SPDES WITH WHITE NOISE FORCING TERMS
Yanzhao Cao
Department of Mathematics & Statistics
Auburn University

November 20, 2009

Yanzhao Cao

Numerical SPDES

November 20, 2009

1 / 35

Outline
1

Introduction
Brownian sheet and elliptic SPDE

Yanzhao Cao

Numerical SPDES

November 20, 2009

2 / 35

Outline
1

Introduction
Brownian sheet and elliptic SPDE

2

SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate

Yanzhao Cao

Numerical SPDES

November 20, 2009

2 / 35

Outline
1

Introduction
Brownian sheet and elliptic SPDE

2

SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate

3

Finite element approximation of elliptic SPDE
Finite element solution Uh

Yanzhao Cao

Numerical SPDES

November 20, 2009

2 / 35

Outline
1

Introduction
Brownian sheet and elliptic SPDE

2

SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate

3

Finite element approximation of elliptic SPDE
Finite element solution Uh

4

Finite element approximations of stochastic Stokes equations Stochastic Stokes Equations

Yanzhao Cao

Numerical SPDES

November 20, 2009

2 / 35

Outline
1

Introduction
Brownian sheet and elliptic SPDE

2

SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate

3

Finite element approximation of elliptic SPDE
Finite element solution Uh

4

Finite element approximations of stochastic Stokes equations Stochastic Stokes Equations

5

Numerical experiments
Numerical experiments
Yanzhao Cao

Numerical SPDES

November 20, 2009

2 / 35

1

Introduction
Brownian sheet and elliptic SPDE

2

SPDE with discretized white noise
˙
Discretization of white noise Wh
Error estimate

3

Finite element approximation of elliptic SPDE
Finite element solution Uh

4

Finite element approximations of stochastic Stokes equations Stochastic Stokes Equations

5

Numerical experiments
Numerical experiments

Yanzhao Cao

Numerical SPDES

November 20, 2009

3 / 35

Brownian sheet and elliptic SPDE
Goal: finite element approximations of SPDES with white
noise forcing terms
(

˙
v(u)(x) = g (x) + (x)W (x);
u(x) = 0; x P @ Ω:

x

P Ω & R d (d = 2; 3);

v: elliptic or Stokes operator

˙
W : White noise
g P L2 (Ω)

Yanzhao Cao

Numerical SPDES

November 20, 2009

4 / 35

Brownian sheet and elliptic SPDE
Brownian Sheet: Generalization of Brownian motion
Brownian sheet W : a random set function such that
1 W (A) $ N (0; jAj); A & Ω, Lebesgue measurable;
(In 1-D, W (t) $ N (0; t))

2 If A ’ B = Y then W (A) and W (B) are independent;
(In 1-D, W (t) has independent increments)

3 If A ’ B = Y thenW (A ‘ B)   W (A) $ N (0; jB j)
(In 1-D W (t + ∆t)   W (t) $ N (0; ∆t))
˙
White noise: W is defined as a distribution:
˙
W () =

Yanzhao Cao

Z



I
dW  P C0 (Ω):

Numerical SPDES

November 20, 2009

5 / 35

Brownian sheet and elliptic SPDE
Stochastic semi-linear elliptic PDE with white noise forcing term
(

˙
 ∆u(x) + f (u(x)) = g (x) + (x)W (x);
u(x) = 0; x P @ Ω:

x

P Ω & R 2;

˙
W : White noise

Yanzhao Cao

Numerical SPDES

November 20, 2009

6 / 35

Brownian sheet and elliptic SPDE
Integral equation formulation: weak solution
Let G = G (x ; y ) be the Green’s function.
K (x) =
˙
K W (x) =

Z



Z



K = ∆ 1

G (x ; y )(y )dy ;

˙
G (x ; y )W (y )dy =

Z


G (x ; y )dW (y ):

Stochastic integral equation:

Z
u + Kf (u) = Kg +


Yanzhao Cao

G (x ; y )dW (y )

Numerical SPDES

November 20, 2009

7 / 35

Brownian sheet and elliptic SPDE
Brief literature survey

 Existence of weak solution:

R. Buckdahn and E. Pardoux, (1990): 2-D and 3-D;
T. Martinez and M. Sanz-Sole (2006): Higher dimension....
Continue Reading

Please join StudyMode to read the full document

You May Also Find These Documents Helpful

  • Essay on Numerical methods
  • numerical methods Essay
  • Gauss & Gauss-Seidel Numerical Methods Essay
  • Numerical Method Essay
  • An Introduction to Programming and Numerical Methods in Matlab Essay
  • Using Numerical Method In Engineering Application Essay
  • numerical computation Essay
  • Essay about Studies on Stress Concentration Using Experimental and Numerical Methods

Become a StudyMode Member

Sign Up - It's Free