# cao's numerical method for solving SPDE

**Topics:**Numerical analysis, Finite element method, Partial differential equation

**Pages:**35 (1985 words)

**Published:**October 11, 2013

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

Ω

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

Ω

Z

Ω

K = ∆ 1

G (x ; y )(y )dy ;

˙

G (x ; y )W (y )dy =

Z

Ω

G (x ; y )dW (y ):

Stochastic integral equation:

Z

u + Kf (u) = Kg +

Ω

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....

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