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SIAM J. NUMER. ANAL.c

2012 Society for Industrial and Applied Mathematics

Vol. 50, No. 6, pp. 2917–2939

TIME-SPLITTING METHODS TO SOLVE THE STOCHASTIC

INCOMPRESSIBLE STOKES EQUATION∗

ERICH CARELLI†, ERIKA HAUSENBLAS‡,AND ANDREAS PROHL†

Abstract. For the stochastic incompressible time-dep endent Stokes equation, we study diﬀerent

time-splitting methods that decouple the computation of velocity and pressure iterates in every

iteration step. Optimal strong convergence is shown for Chorin’s time-splitting scheme in the case of

solenoidal noise, while computational counterexamples show poor convergence behavior in the case

of general stochastic forcing. This suboptimal performance may be traced back to the nonregular

pressure process in the case of general noise. A modiﬁed version of the deterministic time-splitting

method that distinguishes between the deterministic and stochastic pressure removes this deﬁciency,

leading to optimal convergence behavior.

Key words. discretization of stochastic partial diﬀerential equation, splitting scheme, error

analysis

AMS subject classiﬁcations. 60H15, 65M12, 65M60, 76D07

DOI. 10.1137/100819436

1. Introduction. Let (Ω,F,F,P) be a ﬁltered probability space, and let D⊂

Rd,d=2,3, be a bounded polyhedral domain. We consider the d-dimensional stochas-

tic Stokes equation which P-a.s. satisﬁes

ut−Δu+∇p=f+B(·,u)˙

Won DT:= (0,T)×D,

div u=0 onDT,(1.1)

u(0) = u0on D,

u=0on ∂DT:= (0,T)×∂D.

Here, the velocity u=(u1,...,u

d) and the pressure pare unknown random ﬁelds

on DT,andWis an F-adapted cylindrical Wiener process on H

H

H,whereH

H

His a sep-

arable Hilbert space. Possible choices of H

H

Hinclude L2(D)orH1

0(D). Finally, let

t→ Bt, u(t)be an appropriate operator-valued map to be speciﬁed later; as an

example. As an example (see section 6 for details), we may consider a constant op-

erator which maps from H

H

Hto a ﬁnite dimensional subspace of H1

0(D). The study

of the stochastic incompressible Stokes system is, e.g., motivated from modeling mi-

croﬂuids, where inertial eﬀects are generally negligible, and microscopic ﬂuctuations

are relevant contributions to ﬂuid ﬂow dynamics; cf. [18, 7].

Let V

V

V:= {ψ

ψ

ψ∈C∞

0(D): divψ

ψ

ψ=0inD}denote the space of solenoidal func-

tions with closures H:= V

V

VL2(D)and V:= V

V

VH1

0(D). Then, strong solutions u∈

L2(Ω; C([0,T]; H)) ∩L2(ΩT;V), where ΩT:= Ω ×(0,T), of (1.1) for proper operators

Bare usually obtained by a Galerkin method which employs divergence-free approx-

imates from ﬁnite dimensional spaces Hn⊂H(n≥1) to remove the pressure from

the problem. This strategy is diﬀerent from a numerical setting, where the choice of

∗Received by the editors December 28, 2010; accepted for publication (in revised form) August

21, 2012; published electronically November 6, 2012.

http://www.siam.org/journals/sinum/50-6/81943.html

†Mathematisches Institut, Universit¨at T¨ubingen, Auf der Morgenstelle 10, D-72076 T¨ubingen,

Germany (carelli@na.uni-tuebingen.de, prohl@na.uni-tuebingen.de).

‡Lehrstuhl f¨ur Angewandte Mathematik, Montan Universit¨at Leoben, Franz Josef Strasse 18,

A-8700 Leoben, Austria (erika.hausenblas@sbg.ac.at).

2917

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2918 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

the ﬁnite dimensional ansatz space for the pressure, as well as regularity properties of

the pressure from (1.1), crucially determine both stability and convergence behavior

of the resulting scheme; see, e.g., [9].

Properly handling the incompressibility constraint numerically is a nontrivial is-

sue and is usually accomplished in a variational rather than a pointwise sense. As

it is well-known for the corresponding deterministic problem, discretization strategies

based on implicit methods cause a signiﬁcant computational eﬀort due to the coupled

computation of both velocity and pressure iterates. Moreover, choices of stable ﬁnite

element pairings are restricted by the LBB constraint. As a consequence, splitting

algorithms turn out to be a very promising alternative to reducing the complexity

of actual computations by successively updating velocity and pressure iterates; we

refer to [10] for a recent survey on this topic. It is evident that such a strategy is

desirable to solve the stochastic partial diﬀerential equation (1.1), where a signiﬁcant

number of trajectories has to be computed to obtain statistically relevant results for

quantities of interest. The goal of this paper is to show that the interplay of time-

splitting strategies and the “stochastic nature” of problem (1.1) is subtle, leading to a

poor convergence behavior of known time-splitting schemes which perform well in the

deterministic case. Computational experiments detail this assertion, which is rooted

in the nonregular pressure process in (1.1). In a second step, an optimally convergent

stochastic time-splitting scheme is constructed that distinguishes between approxi-

mations of the (nonregular) stochastic pressure and the (more regular) deterministic

pressure.

To illustrate the problematic issue to construct a proper time-splitting scheme for

a stochastic equation, we start with Chorin’s pro jection method [4, 6, 19], which is

one of the ﬁrst splitting schemes to solve the deterministic incompressible (Navier–)

Stokes equation. Consider the ﬁltered probability space (Ω,F,F,P). Let fm+1 :=

f(tm+1,·)∈L2(Ω,L2), suppose that u0∈L2(Ω,V) is given, and consider inde-

pendently and identially distributed stochastic increments ΔWm+1 := W(tm+1)−

W(tm), where k=tm+1 −tm>0 denotes the mesh-size of the equidistant grid

Ik:= {tm}M

m=0 covering [0,T].

Algorithm 1.1. 1. Let m≥0. For given um∈L2(Ω,V)and

um∈L2(Ω,H1

0(D)),

ﬁnd

um+1 ∈L2(Ω,H1

0(D)) such that P-a.s.

(1.2)

um+1 −um−kΔ

um+1 =kfm+1 +B(tm,

um)ΔWm+1 on D.

2. Compute um+1 ∈L2(Ω,H)andpm+1 ∈L2(Ω,H1(D)/R) such that P-a.s.

um+1 −

um+1 +k∇pm+1 =0,div um+1 =0 onD,(1.3)

um+1,n=0 on∂D.(1.4)

We start a discussion of the scheme which ignores the stochastic term for a mo-

ment: the latter step can be reformulated as a problem for the pressure function

only,

(1.5) −Δpm+1 =−1

kdiv

um+1 on D, ∂npm+1 =0 on∂D.

Hence, each step consists of (1.2), (1.5), and the algebraic update (1.3) to obtain

um+1 ∈H.

In order to understand error eﬀects inherent to discretization in time, and operator

splitting in Chorin’s scheme, we shift the index in (1.3)1back and add the resulting

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THE STOCHASTIC STOKES EQUATION 2919

equation to (1.2); together with (1.5), we then arrive at

um+1 −

um−kΔ

um+1 +k∇pm=kfm+1 +B(tm,

um)ΔWm+1 on D,(1.6)

div

um+1 −kΔpm+1 =0 onD,(1.7)

∂npm+1 =0 on∂D,(1.8)

and

u0≡u0on D. We make the following observations: (i) iterates {

um}m≥0

of Algorithm 1.1 are no longer divergence-free but satisfy the “quasi-compressibility

equation” (1.7) with a penalization parameter equal to k; (ii) iterates of the pressure

satisfy a homogeneous Neumann boundary condition, which is in contrast to pressure

p:DT→Rfrom (1.1); and (iii) the pressure iterate in (1.6) is used in an explicit

fashion, which rules out an immediate discrete energy law, where test functions um+1

and pm+1 are used.

For the deterministic case, by assuming D⊂Rdto be a convex polyhedral domain,

u0∈V∩H2(D), and f∈W2,∞(0,T;L2(D)), the following optimal estimates are

proved in [16, Theorem 6.1]:

(1.9) max

1≤m≤M√τmu(tm,·)−

umL2+√ku(tm,·)−

umH1≤Ck,

where τm:= min{1,t

m}. Its proof consists of three steps. First, optimal error

estimates for the implicit Euler discretization using solenoidal velocity ﬁelds are

derived, where its derivation beneﬁts from valid regularity properties of solutions

u∈C([0,T]; V∩H2)∩H2(0,T;V), where Xdenotes the dual of the Banach space

X. Then, a modiﬁed version of (1.6)–(1.8) is studied with respect to both conver-

gence and stability properties, where the pressure iterate pmin (1.6) is shifted to

pm+1; a key property here is the existing bound p∈L∞(0,T;H1/R) for the deter-

ministic evolutionary incompressible Stokes problem. We remark that this pressure-

stabilization method (with parameter ε=k) is of its own interest, since it allows

for more choices of ﬁnite element pairings [2, 12], which are usually restricted by

the discrete LBB condition. Finally, the third step accounts for the explicit treat-

ment of the pressure in (1.6), which strongly beneﬁts from an upper bound for

T

0τ(s)∇pt(s)2dsfor the pressure from (1.1) in terms of the data u0,f,andDT,

where τ(s)=min{1,s}.

The goal of the present work is to study convergence properties of H1

0(D)-valued

iterates {

um}mfrom Algorithm 1.1 to approximate solutions of (1.1). The main diﬃ-

culties which enter in the stochastic setting are due to restricted regularity properties

(in time) of solutions (u,p) to (1.1), which are due to the driving stochastic term;

for instance, the pressure which is constructed by Helmholtz decomposition after uis

found need not even be absolutely continuous with respect to time [13] (see (2.7)), but

its regularity properties are crucial for the convergence analysis of our splitting method

as detailed above. Hence, there is the question of whether splitting eﬀects inherent

to Algorithm 1.1 will deteriorate convergence rates of computed iterates {

um}m—

if compared to divergence-free velocity iterates {wm}m⊂L2(Ω; V), approximating

{u(tm,·)}m, and solving the coupled Euler–Maruyama time discretization of (1.1),

which holds P-a.s.,

(1.10)

wm+1 −wm−kΔwm+1 +k∇qm+1 =kfm+1 +B(tm,wm)ΔWm+1 on D,

div wm+1 =0 onD,(1.11)

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2920 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

where P-a.s. w0=u0on D. Note that the pressure qm+1 :Ω×D→R,whichap-

proximates p(tm,·), will be eliminated from the convergence analysis when solenoidal

test functions are used. As a consequence, the following rates of strong convergence

of Euler iterates {wm}mare proved in [11]:

max

1≤m≤MEu(tm,·)−wm2

L21/2

(1.12)

+Ek

1≤m≤M∇u(tm,·)−wm2

L21/2≤CT√k.

In fact, [11, Theorem 3.1] provides rates of convergence for a ﬁnite dimensional Wiener

process. However, the proof can be modiﬁed in such a way that the same result holds

for a Wiener process having a covariance operator with ﬁnite trace. Since this will be

the case (see Assumption (2.3) below), Theorem 3.1 of [11] is applicable.

The ﬁrst main result in this paper is Theorem 3.1, which shows property (1.12)

for iterates {

um}from Algorithm 1.1 in the case of solenoidal noise. A discretiza-

tion in space using equal-order ﬁnite elements is studied in section 5, and overall

error estimates for related ﬁnite element iterates {

um}are given in Theorem 5.1.

Then, computational studies are provided in section 6 which compare convergence

behavior of iterates from Algorithm 1.1 and (1.10)–(1.11) for solenoidal and general

L2-noise and highlight that solenoidal noise is imperative for optimal convergence

behavior of the splitting Algorithm 1.1, which in the case of general noise deterio-

rates to a poor convergence behavior. Those computational studies motivate the new

time-splitting scheme (Algorithm 4.1) in section 4, which distinguishes between ap-

proximate deterministic and stochastic pressure iterates. As a consequence, optimal

rate of convergence for general noise is shown both theoretically (see Theorem 4.1)

and computationally.

The remainder of this work is organized as follows. Necessary background for

the stochastic partial diﬀerential equation (1.1) and useful stability bounds for Eu-

ler iterates {wm}msolving (1.10)–(1.11) are provided in section 2. In section 3, we

estimate the additional diﬀerent perturbation eﬀects due to the quasi-compressibility

constraint (1.7) and the splitting character of Algorithm 1.1 due to the explicit treat-

ment of the pressure in (1.6), which then leads to Theorem 3.1. In section 5, a ﬁnite

element discretization of Algorithm 1.1 is proposed, where the study of the coupled er-

ror eﬀects due to time discretization, time splitting, and spatial discretization leads to

Theorem 5.1. Computational evidence to highlight the failure of Chorin’s method in

the case of general noise is reported in section 6, as well as the modiﬁed Algorithm 4.1

that performs optimally for general noise.

2. Preliminaries. For a d om a i n D⊂Rd,d=2,3, let L2(D) denote the usual

Lebesgue space of square integrable functions, endowed with the scalar product (·,·)

and norm ·

L2. We employ the usual Sobolev spaces Wk,p(D)andW1,p

0(D) with

norm ·

Wk,p , while for p= 2 we use the notation Hk(D)=Wk,2(D). Since we

always work on D, most of the time the letter Dwill be dropped. For vector-valued

functions, the corresponding spaces are denoted by bold letters, e.g., (H1)d=H1.

2.1. The problem. Let (Ω,F,F,P) be a complete probability space with con-

tinuous ﬁltration F={Ft;t≥0}, and let {βj(t); t≥0}j,j∈N, be a sequence of

independent identically distributed R-valued Brownian motions on (Ω,F,F,P). Let

H

H

Hbe a Hilbert space and {ej;j=1,2,...}be an orthonormal basis of H

H

H.Wedenote

by W={W(t); t≥0}the cylindrical Wiener process on (Ω,F,F,P), which is deﬁned

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

THE STOCHASTIC STOKES EQUATION 2921

as

[0,∞)t→ W(t)= ∞

j=1

βj(t)ej.

For p≥1andK

K

Kbeing a Hilbert space, we denote by Mp

F(0,T;K

K

K)(:= Mp

F(Ω ×

[0,T]; K

K

K)) the space of all F-adapted processes belonging to Lp(Ω ×[0,T]; K

K

K). More-

over, let L2(H

H

H,K

K

K) denote the space of linear operators from H

H

Hto K

K

Khaving a ﬁnite

Hilbert–Schmidt norm. For any process ϕ

ϕ

ϕ∈M2

F(0,T;L2(H

H

H,K

K

K)) we deﬁne the stochas-

tic integral t

0ϕ

ϕ

ϕ(s)dW(s), 0 ≤t≤T, as the unique continuous K

K

K-valued F-martingale

such that for all h∈K

K

Kit holds that

t

0

ϕ

ϕ

ϕ(s)dW(s),hK

K

K

= lim

J→∞

J

j=1 t

0ϕ

ϕ

ϕ(s)ej,hK

K

Kdβj(s)∀0≤t≤T.

Moreover, its second moment satisﬁes Ito’s isometry property,

E⎡

⎣

T

0

ϕ

ϕ

ϕ(s)dW(s)

2

K

K

K⎤

⎦=ET

0ϕ

ϕ

ϕ(s)2

L2(H

H

H,K

K

K)ds.

Remark 2.1. This construction includes the case of a Q-Wiener process with

values in H

H

Hand a positive self-adjoint covariance operator Q:H

H

H→H

H

Hof trace class,

i.e., with eigenvalues {λj}∞

j=1 satisfying ∞

j=1 λj<∞, and eigenfunctions {ej}∞

j=1 ⊂

H

H

Hwhich build an orthonormal basis. Then we may represent a Q-Wiener process

with covariance operator Qin the form

[0,∞)t→ W(t)= ∞

j=1 λjβj(t)ej.

Thus, Ito’s isometry reads

E⎡

⎣

T

0

ϕ

ϕ

ϕ(s)dW(s)

2

K

K

K⎤

⎦=ET

0ϕ

ϕ

ϕ(s)◦Q1/22

L2(H

H

H,K

K

K)ds.

We recover the stochastic integral with the cylindrical Wiener process if we set

Q=Id (which has no ﬁnite trace). For such a Q-Wiener process it is possible to

enlarge the class of integrands to the space M2

F(0,T;L2(Q1/2(H

H

H),K

K

K)), which includes

M2

F(0,T;L2(H

H

H,K

K

K)).

Recall the Stokes operator A≡−PHΔ with domain D(A)=V∩H2(D), which

is endowed with the norm ·

D(A)=A·

L2. Here, PH:L2→Hdenotes the

(Leray) projection operator. For Lipschitz domains D⊂Rdand on V∩H2(D), the

operator norm ·D(A)is equivalent to the H2(D)-norm. Throughout this work, we

assume that the domain Dis such that for the solution of the Stokes equations

−Δu+∇π=fon D, div u=0 onD, u=0 on∂D,

there holds the bound

uH2+πH1≤CfL2

(2.1)

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2922 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

if f∈L2. In two dimensions this is known to be true for convex polygonal domains,

while in three dimensions this holds for C2boundaries (see [20, Proposition 2.2]), and

it is believed to hold for convex polyhedra as well. Throughout the paper, let

(2.2) u0∈L2Ω,F0,P;Vand f∈L2Ω×(0,T); L2(D).

Suppose that B:[0,T]×H1

0→L

2(H

H

H;K

K

K) is measurable, Lipschitz, and sublinear for

K

K

K=L2and K

K

K=H1

0; more precisely, there exists a constant CT>0 such that for

K

K

K=L2and K

K

K=H1

0,

B(t, v)−B(t, w)

L2(H

H

H,K

K

K)≤CTv−wK

K

K∀t∈[0,T]∀v,w∈H1

0,(2.3)

and for v∈H1

0

B(·,v)∈L2Ω; L20,T;L2(H

H

H,H1

0).(2.4)

We cal l a n F-adapted stochastic process a strong solution of (1.1) (in the stochastic

sense) if u∈L2(Ω; C([0,T]; H))∩M2

F(0,T;V) such that for all t∈[0,T]andallψ

ψ

ψ∈V

there holds P-a.s.,

u(t),ψ

ψ

ψ+t

0∇u(s),∇ψ

ψ

ψds=u0,ψ

ψ

ψ+t

0f(s),ψ

ψ

ψds+t

0

B(s, u)dW(s),ψ

ψ

ψ.

The existence of a unique strong solution u∈L2(Ω; C([0,T]; V)) ∩M2

F(0,T;D(A))

which satisﬁes P-a.s. the energy equation

u(t)2

L2+2t

0∇u(s)2

L2ds

=u02

L2+2t

0f(s),u(s)ds+2t

0Bs, u(s)dW(s),u(s)

(2.5)

+t

0Bs, u(s)2

L2(H

H

H;L2)ds∀t∈[0,T]

is well-known; see, for instance, [15, Theorem 6.19]. Moreover, standard arguments

yield for u0∈L2(Ω,F0,P;V)thatu∈L2(Ω; C([0,T]; V)) ∩M2

F(0,T;D(A)), and

Esup

0≤t≤Tu(t)2

H1+ET

0Au(s)2

L2ds

≤CT1+Eu02

H1+ET

0f(s)2

L2ds

+ET

0B(s, 0)2

L2(H

H

H,L2)ds;(2.6)

see, e.g., [5, Theorem 4.4 and section 5]. In contrast, the limited available ana-

lytical results about the pressure in (1.1) indicate very restricted smoothness. For

B:[0,T]×L2(D)→L

2(H

H

H,L2(D)), there exists a unique (distributional) pressure

(see [13, Theorem 2.2])

(2.7) p∈L1Ω,F,P;W−1,∞0,T;L2(D)

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THE STOCHASTIC STOKES EQUATION 2923

such that P-a.s.

ut−Δu+∇p=f+B(·,u)˙

Win D(DT)d,(2.8) D

pdx=0 inD(0,T).(2.9)

This result gives evidence of a deregularizing eﬀect upon the pressure in (1.1) which

is exerted by a general noise. This feedback eﬀect of general noise onto the (lack

of) regularity of the pressure may be avoided by analytical constructions using Leray

projection but causes severe deterioration with respect to accuracy of well-known

numerical schemes where accurate pressures are needed.

As will be shown in Lemma 2.1 below, pressure iterates of the (coupled) Euler–

Maruyama scheme (1.10)–(1.11) are more regular for noise that is solenoidal, which

is why we assume

(2.10) B:[0,T]×H1

0(D)→L

2(H

H

H,V)

in sections 3 and 5. Conversely, computational experiments in section 6.1 show that

Chorin’s projection method performs optimally only in the case of solenoidal noise.

In the case B:[0,T]×H1

0→L

2H

H

H;V, related a priori estimates in Lemma 2.1

motivate

(2.11) p∈L1Ω,F,P;L2(0,T;H1(D)/R),

which provides enough regularity of the pressure such that the splitting scheme per-

forms optimally. However, we are not aware of a rigorous analytical motivation of

this assertion for the limiting equations (1.1).

Remark 2.2. A velocity ﬁeld uthat solves the stochastic incompressible (Navier–)

Stokes equations is usually constructed by an (“inner approximation”) Galerkin method

that employs solenoidal test functions and thus eliminates the pressure pfrom the

problem in a ﬁrst step; a pressure pis then later obtained by de Rham’s theorem;

see, e.g., [1, 8, 5, 13]. A diﬀerent strategy is to obtain solutions by perturbing the in-

compressibility constraint (“quasi-compressibility method”) to avoid the saddle-point

character of the problem, for example, (ε>0):

(i) div uε+εpε=0 onDT,

(ii) div uε−εΔpε=0 onDT,∂

npε=0 on∂DT,

(iii) div uε+εpε

t=0 onDT,p

ε(0) = p(0) on D,

(iv) div uε−εΔpε

t=0 onDT,∂

npε=0 on∂DT,p

ε(0) = p(0) on D.

The penalty method (i) is used in [3], and the artiﬁcial compressibility method (iii) in

[14] to construct solutions of the stochastic incompressible Navier–Stokes equations.

The pressure stabilization ansatz (ii) is related to Algorithm 1.1, where ε=kis chosen

in (1.5). The pressure correction method (iv) is used for numerical schemes as well;

cf. [16] for further details.

2.2. Euler scheme. Suppose that (2.2)–(2.4) and (2.10) are valid throughout

the section. For every m≥0, there exists a solution wm+1 ∈L2Ω; Vsuch that

w0=u0and P-a.s.

(wm+1 −wm,ϕ

ϕ

ϕ)+k(∇wm+1,∇ϕ

ϕ

ϕ)

=k(fm+1,ϕ

ϕ

ϕ)+B(tm,wm)ΔWm+1,ϕ

ϕ

ϕ∀ϕ

ϕ

ϕ∈V.(2.12)

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2924 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

Both P-a.s. existence and uniqueness of iterates from (1.10)–(1.11) follow by V-

coercivity of the bilinear form related to the Stokes equations. Moreover, solutions

satisfy the error estimate given in (1.12) and shown in [11].

Some bounds for solutions of (1.10)–(1.11) in strong norms will be useful later,

where the ﬁrst one mimics (2.5) on a discrete level.

Lemma 2.1. Let {wm}m≥1⊂L2Ω; Vbe a soluti on of (2.12),andlet(2.2),

(2.3),(2.4),and(2.10) be valid. Then

(i) max

1≤m≤M

Ewm2

L2+EM

m=1 wm−wm−12

L2+Ek

M

m=1 ∇wm2

L2

≤CTEu02

L2+Ek

M

m=1 fm2

L2,

(ii) max

1≤m≤M

E∇wm2

L2+EM

m=1 ∇[wm−wm−1]2

L2+Ek

M

m=1 Awm2

L2

≤CTE∇u02

L2+Ek

M

m=1 fm2

L2,

(iii) Ek

M

m=1 ∇qm2

L2≤CTE∇u02

L2+Ek

M

m=1 fm2

L2,

where CT≡C(u0,B,f,D,T)>0is a generic constant that does not depend on k.

Proof.Assertion (i). Choose ϕ

ϕ

ϕ=wm+1 in (2.12), and use the algebraic identity

2a−b,a=|a|2−|b|2+|a−b|2to obtain

1

2wm+12

L2−wm2

L2+wm+1 −wm2

L2+k∇wm+12

L2

=kfm+1,wm+1 +B(tm,wm)ΔWm+1,wm+1 −wm

(2.13)

+B(tm,wm)ΔWm+1,wm.

Taking expectations puts the last term in (2.13) to zero. For the remaining stochastic

term, we use the Ito isometry and (2.3), (2.4) to conclude that

EB(tm,wm)ΔWm+1,wm+1 −wm

≤kEB(tm,wm)2

L2(H

H

H,L2)+1

4Ewm+1 −wm2

L2

≤CTk1+Ewm2

L2+1

4Ewm+1 −wm2

L2.

We now use the discrete version of Gronwall’s lemma in (2.13) to obtain assertion (i).

Assertion (ii). Formally take ϕ

ϕ

ϕ=Awm+1 in (2.12), and proceed as before. We

use (2.10) and integrate by parts in the stochastic term to ﬁnd

∇B(tm,wm)ΔWm+1,∇wm

(2.14)

+∇B(tm,wm)ΔWm+1,∇wm+1 −wm.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

THE STOCHASTIC STOKES EQUATION 2925

After taking expections, only the second term is nonzero; by Ito’s isometry, (2.3), and

(2.4), an upper bound for it is

1

4E∇[wm+1 −wm]2

L2+CkEB(tm,wm)2

L2(H

H

H,H1)

≤1

4E∇[wm+1 −wm]2

L2+CTk1+Ewm2

H1.

Putting things together and using the discrete Gronwall inequality then leads to

assertion (ii).

Assertion (iii). For every m≥0, consider (2.12) in strong form on L2(Ω; L2),

which is justiﬁed from the previous step. Termwise multiplication with ∇qm+1 and

integration in space then leads to

k

2∇qm+12

L2≤CkΔwm+12

L2+fm+12

L2,

where we use (2.10). Assertion (ii) and (2.1) then validate the claim.

3. Perturbation eﬀects in Algorithm 1.1: Quasi-compressibility and

operator-splitting. Solutions {

um}m⊂L2Ω; H1

0(D)of Algorithm 1.1 satisfy

(1.6)–(1.8), which illustrates the diﬀerent error eﬀects due to time discretization,

quasi-incompressibility, and splitting character in the scheme. The main result of this

section is the following theorem.

Theorem 3.1. Let T>0and D⊂Rd,d=2,3, be such that (2.1) holds, and let

(2.2)–(2.4), (2.10) be valid. Denote by u∈L2(Ω; C([0,T]; V)) ∩L2(Ω; L2(0,T;H2))

the strong solution of (1.1),and{

um}m⊂L2(Ω; H1

0(D)) solves Algorithm 1.1.There

exists a constant C≡C(E[u02

H1],D

T)>0such that

(3.1)

max

1≤m≤MEu(tm,·)−

um2

L21/2+ Ek

M

m=1 ∇u(tm,·)−

um2

L2!1/2

≤C√k.

The proof is split into several steps: ﬁrst, we study solutions {(vm,r

m)}m⊂

L2(Ω; H1

0(D)) ×L2(Ω,H1(D)/R) of an auxiliary problem, where P-a.s.

vm+1 −vm−kΔvm+1 +k∇rm+1 =kfm+1 +B(tm,vm)ΔWm+1 on D,

div vm+1 −kΔrm+1 =0 onD,(3.2)

∂nrm+1 =0 on∂D,

and v0≡u0on D. Note that in contrast to (1.6), where the approximation of the

pressure is given from the previous time-step, it is here computed by an implicit

procedure. Our goal is to show both convergence of iterates {vm,r

m}mtoward the

solution of (1.10) and stability behavior. Then, we study convergence behavior for

solutions of (3.2) to that of (1.6)–(1.8).

Proof.Step 1. The pressure stabilization problem (3.2): Rates of convergence. We

show the following convergence estimate for solutions {wm}m⊂L2(Ω; V) of (2.12)

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2926 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

and {vm}m⊂L2(Ω,H1

0(D)) of (3.2):

max

1≤m≤MEwm−vm2

L21/2+⎛

⎝E⎡

⎣k

1≤m≤M∇wm−vm2

L2⎤

⎦⎞

⎠

1/2

+⎛

⎝E⎡

⎣k2

1≤m≤M∇qm−rm2

L2⎤

⎦⎞

⎠

1/2

≤C√k.(3.3)

Let em:= wm−vm∈L2(Ω,H1(D)) and χm:= qm−rm∈L2(Ω,H1(D)/R). Taking

the diﬀerence of (1.10) and (3.2) then leads to P-a.s.

em+1 −em−kΔem+1 +k∇χm+1 =B(tm,wm)−B(tm,vm)ΔWm+1 on D,

div em+1 −kΔχm+1 =−kΔqm+1 on D,

∂nχm+1 =∂nqm+1 on ∂D,(3.4)

and e0≡0in D. By testing the ﬁrst equation with em+1 and using Lipschitz

continuity of Band testing the second with χm+1 and using (3.4)3for integration by

parts, adding both identities and using Young’s inequality then leads to P-a.s.

1

2em+12

L2−em2

L2+em+1 −em2

L2+k∇em+12

L2+k2∇χm+12

L2

≤[B(tm,wm)−B(tm,vm)]ΔWm+1,em+1

4em−em−12

L2

+[B(tm,wm)−B(tm,vm)]ΔWm+12

L2+1

4k2∇χm+12

L2+k2∇qm+12

L2.

The leading term on the right-hand side vanishes when we take its expectation. By

Ito’s isometry, and (2.3), there holds for the remaining stochastic integral term

E[B(tm,wm)−B(tm,vm)]ΔWm+12

L2≤Ck1+Eem2

L2.

We now take expectation termwise and sum over all steps 0 ≤m≤m∗≤M−1;

because of E[e02

L2] = 0, Lemma 2.1(iii), and the discrete version of Gronwall’s

inequality, after summation we arrive at

1

2Eem∗+12

L2+1

4Em∗

m=0 em+1 −em2

L2+Ek

m∗

m=0 ∇em+12

L2

+3

4Ek2

m∗

m=0 ∇χm+12

L2≤Ctm∗Ek2

m∗

m=0 ∇qm+12

L2≤CTk.(3.5)

Step 2. The pressure stabilization problem (3.2): Stability. Proper bounds are

needed for the pressure in (3.2) to validate optimal error estimates between solutions

of (3.2) and (1.6)–(1.8) below. We show

(3.6) max

1≤m≤M

Evm2

H1+Ek

M

m=1 vm2

H2+Ek

M

m=1 ∇rm2

L2≤CT.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

THE STOCHASTIC STOKES EQUATION 2927

Hence, for solutions of problem (3.2) there hold the same estimates which are valid

for solutions of (2.12) provided in Lemma 2.1.

Property (3.6)1follows from the term (3.3)3and Lemma 2.1(ii), and property

(3.6)3is a consequence of (3.3)3and Lemma 2.1(iii). A formal derivation of (3.6)2

uses (3.2)1, which we multiply by −Δvm+1 and then integrate over D. After summing

up over all 0 ≤m≤M−1, by taking expectations and absorbing terms we arrive at

1

2E∇vM2

L2+1

2EM−1

m=0 ∇(vm+1 −vm)2

L2+1

4Ek

M−1

m=0 Δvm+12

L2

≤1

2E∇v02

L2+CEk

M−1

m=0 ∇rm+12

L2+CEk

M−1

m=0 fm+12

L2

(3.7)

+kEM−1

m=0 B(tm,vm)2

L2(H

H

H,H1)+1

4EM−1

m=0 ∇[vm+1 −vm]2

L2,

whereweusethefactthatE[M

m=1(∇B(tm,vm)ΔWm+1 ,∇vm)] = 0 and Ito’s isome-

try. By (2.3), (2.4), we have E[B(tm,vm)L2(H

H

H,H1)]≤CT(1+ vmH1). The bounds

(3.6)1,3then allow us to conclude (3.6)2from (3.7) after using the discrete version of

Gronwall’s inequality.

Step 3. The splitting error: Comparison of problems (3.2) and (1.6)–(1.8). We

estimate the diﬀerences ε

ε

εm:= vm−

um∈L2Ω,H1

0(D)and ηm:= χm−pm∈

L2Ω,H1(D)/R, which are determined by the following system of equations, which

hold P-a.s.:

ε

ε

εm+1 −ε

ε

εm−kΔε

ε

εm+1 +k∇ηm=E

E

Em+1 on D,

div ε

ε

εm+1 −kΔηm+1 =0 onD,(3.8)

∂nηm+1 =0 on∂D,

where ε

ε

ε0≡0,and

(3.9) E

E

Em+1 := −k∇[rm+1 −rm]+B(tm,vm)−B(tm,

um)ΔWm+1.

Upon testing (3.8)1by ε

ε

εm+1 and (3.8)2by ηm+1, adding both identities, using Young’s

inequality with δ1>0, and absorbing terms then yields

1

2ε

ε

εm+12

L2−ε

ε

εm2

L2+ε

ε

εm+1 −ε

ε

εm2

L2+k∇ε

ε

εm+12

L2+k2∇ηm+1,∇ηm

≤[B(tm,vm)−B(tm,

um)]ΔWm+1,ε

ε

εm−k∇(rm+1 −rm),ε

ε

εm+1

+Cδ1[B(tm,vm)−B(tm,

um)]ΔWm+12

L2+δ1ε

ε

εm+1 −ε

ε

εm2

L2.(3.10)

Again, the expectation of the leading term on the right-hand side vanishes; Ito’s

isometry and (2.3) then imply

E[B(tm,vm)−B(tm,

um)]ΔWm+12

L2≤CTk1+E[ε

ε

εm2

L2].

It remains to deal with terms which contain pressures. We use (3.8)2and Young’s

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2928 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

inequality with δ2>0 to conclude that

k2∇ηm+1,∇ηm=k2∇ηm+1 2

L2−k2∇ηm+1,∇[ηm+1 −ηm]

=k2∇ηm+12

L2−k∇ηm+1,ε

ε

εm+1 −ε

ε

εm

(3.11)

≥1−1

4δ2k2∇ηm+12

L2−δ2ε

ε

εm+1 −ε

ε

εm2

L2.

The remaining crucial term in (3.10) is bounded as follows:

k2∇[rm+1 −rm],∇ηm+1≤k2δ3∇ηm+1 2

L2+k2

4δ3∇[rm+1 −rm]2

L2,

where we used Young’s inequality with δ3>0. To keep the corresponding terms in

(3.12) nonnegative, we choose parameters δi>0, i=1,2,3, such that

1−1

4δ2−δ3≥0and1

2−δ1−δ2≥0.

Next, we sum over all 0 ≤m≤m∗≤M−1 in (3.10) and take expectations. Then,

by the discrete version of Gronwall’s inequality,

Eε

ε

εm∗+12

L2+1

2−δ1−δ2Em∗

m=0 ε

ε

εm+1 −ε

ε

εm2

L2+Ek

m∗

m=0 ∇ε

ε

εm+12

L2

+k21−1

4δ2−δ3Em∗

m=0 ∇ηm+12

L2≤CT4k2

4δ3

m∗+1

m=0 ∇rm2

L2≤CtTk,(3.12)

where the last estimate uses (3.6)3and r0≡0 is a consistent choice from of (3.2)2,3.

Putting together results (1.12), (3.3), and (3.12) yields the error bound

max

1≤m≤MEu(tm,·)−

um2

L21/2

+⎛

⎝E⎡

⎣k

1≤m≤M∇u(tm,·)−

um2

L2⎤

⎦⎞

⎠

1/2

≤C√k+√k+√k,(3.13)

which proves Theorem 3.1.

The following stability result for solutions of Algorithm 1.1 will be helpful in

section 5, where we consider an optimally convergent, practical ﬁnite element dis-

cretization.

Lemma 3.1. Let T>0,D⊂Rd,d=2,3, be such that (2.1) holds, and let (2.2)–

(2.4),(2.10) be valid. Let {

um}m≥1⊂L2(Ω,H1

0(D)) be the solution of Algorithm 1.1.

Then, all estimates given in Lemma 2.1 remain valid.

Proof. First, we observe that since (1.6) requires solving a linear elliptic boundary

valueproblem,wehave˜

um∈L2(Ω; H2(D)∩H1

0(D)). Then we use (3.6), together

with (3.12), to validate bounds (i), (iii), and (ii)1,2in Lemma 2.1 for {

um}.Inorderto

(formally) verify E[kM

m=1 Δ

um2

L2]≤C, we multiply (1.6) by −Δ

um+1,integrate

over D, and consider expectations. Similar arguments as above lead to

1

2E&∇

um+12

L2−∇

um2

L2+3

4∇[

um+1 −

um]2

L2'+3k

4EΔ

um+12

L2

≤Ek∇pm2

L2+Ck1+E

um2

H1+CkEfm+12

L2.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

THE STOCHASTIC STOKES EQUATION 2929

We now sum up 0 ≤m≤Mand may use the available bound E[kM

m=1 ∇pm2

L2]≤

Cto obtain

Ek

M

m=1 Δ

um2

L2≤CT.

4. Chorin scheme with stochastic pressure correction. As has been shown

so far, we can prove optimal convergence behavior for Algorithm 1.1 only in the

case of solenoidal noise. Here we try to modify Algorithm 1.1 in order to validate

optimal convergence behavior also in the case that the sequence of random variables

{B(tm,

um)ΔWm+1}m≥0approximates general noise. The scheme that we propose

is the following.

Algorithm 4.1. Let m≥0.

1. For given

um∈L2(Ω,H1

0(D)), ﬁnd ξ

ξ

ξm+1 ∈L2(Ω,H) such that P-a.s.

ξ

ξ

ξm+1 +∇sm+1 =1

√kB(tm,

um)ΔWm+1 on D,

div ξ

ξ

ξm+1 =0 onD,(4.1)

ξ

ξ

ξm+1,n=0 on∂D.

2. For given um∈L2(Ω,H), ﬁnd

um+1 ∈L2(Ω,H1

0(D)) such that P-a.s.

(4.2)

um+1 −um−kΔ

um+1 =kfm+1 +√kξ

ξ

ξm+1 on D.

3. Compute um+1 ∈L2(Ω,H)andpm+1 ∈L2(Ω,H1(D)/R) from the following

equations, which hold P-a.s.:

um+1 −

um+1 +k∇pm+1 =0 onD,

div um+1 =0 onD,(4.3)

um+1,n=0 on∂D.

4. Compute the approximation of the pressure pvia

rm+1 =pm+1 +1

√ksm+1.

The underlying idea for this algorithm is to distinguish between deterministic and

stochastic (forcing) terms on the right-hand side of (1.1), which scale diﬀerently in a

time-discretization scheme. Corresponding Helmholtz decompositions of both terms

involve gradient functions, which are then referred to as deterministic and stochastic

pressures. It is by step 1 that the gradient of the stochastic pressure {sm}m≥1in

(4.1)1(which is the Lagrange multiplier resulting from the Leray projection) has no

inﬂuence on computing velocity iterates in steps 1 to 3, where only the deterministic

pressure {pm}m≥1is involved. This argument is further detailed by the following

formal computation for Euler iterates from (1.10)–(1.11):

wm+1 −kΔwm+1 +k∇pm+1 =wm+kfm+1 +B(tm,wm)ΔWm+1

=wm+kfm+1 +√kξ

ξ

ξm+1 +√k∇sm+1,

where ξ

ξ

ξm+1 =1

√kPH[B(tm,wm)ΔWm+1]. As a consequence, we get

wm+1 −kΔwm+1 +k∇πm+1 =wm+kfm+1 +PHB(tm,wm)ΔWm+1,

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

2930 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

where

πm+1 =pm+1 −1

√ksm+1.

In fact, Algorithm 4.1 is Algorithm 1.1, which is applied to the same equation with pro-

jected noise. So, the proof of the convergence rate follows directly from Theorem 3.1.

Unfortunately, for v∈H1

0(D) the projection PHv∈H1(D) is not an element of

H1

0(D). As a consequence, in formula (2.14) we obtain an additional boundary in-

tegral which is diﬃcult to bound, and properties (ii) and (iii) of Lemma 2.1 are not

evident to remain valid in this setting anymore. To avoid this problematic issue, we

consider Problem 1.1 with space periodic boundary conditions on a set Q=[0,L]d,

L>0. Let Hper ,Vper ,andHn

per denote the space periodic analogues of the spaces

H,V,andHn. In this case optimal convergence rates of the splitting Algorithm 4.1

also hold for general noise. We have the following theorem.

Theorem 4.1. Let T>0,D⊂Rd,d=2,3, be such that (2.1) holds, and let

(2.2)–(2.4) be valid. Denote by u∈L2(Ω; C([0,T]; Vper)) ∩L2(Ω; L2(0,T;H2

per)) the

strong solution of (1.1),and{

um}m⊂L2(Ω,H1

per (D)) solves Algorithm 4.1. There

exists a constant C≡C(E[u0H1],D

T)>0such that

(4.4)

max

1≤m≤MEu(tm,·)−

um2

L21/2+ Ek

M

m=1 ∇u(tm,·)−

um2

L2!1/2

≤C√k.

Again, note that condition (2.10) is not needed in this case to validate (4.4).

5. Finite element discretization of Algorithm 1.1. Let Thbe a quasi-

uniform triangulation of the polygonal or polyhedral bounded Lipschitz domain D⊂

Rdinto triangles or tetrahedra for d=2ord= 3, respectively. We deﬁne the lowest

order ﬁnite element space

Hh=(Φ∈C(D): Φ

K∈P

1(K)∀K∈T

h),

where P1(K) denotes the set of polynomials of degree less than or equal to one if

restricted to the element K∈T

h. We introduce equal-order ﬁnite element function

spaces

Hh:= Hhdand Lh:= Hh∩L2(D)/R

and H0

h:= Hh∩H1

0(D). We recall the L2-orthogonal projection P0

h:L2→H0

h,

where

(φ

φ

φ−P0

hφ

φ

φ, ξ

ξ

ξ=0 ∀ξ

ξ

ξ∈H0

h,

for which holds

φ

φ

φ−P0

hφ

φ

φL2+h∇(φ

φ

φ−P0

hφ

φ

φ)L2≤Ch2φ

φ

φH2∀φ

φ

φ∈H2.

Accordingly, there holds for P1

h:H1(D)/R→Lh,where

∇[χ−P1

hχ],∇η=0 ∀η∈Lh,

that

χ−P1

hχL2+h∇[χ−P1

hχ]L2≤Ch2χH2∀χ∈H1/R∩H2.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

THE STOCHASTIC STOKES EQUATION 2931

Below we use ﬁnite elements for a fully discrete version of Algorithm 1.1. More-

over, for simplicity we assume that Bis independent of time.

Algorithm 5.1. Let m≥0.Set

˜

U0:= U0for U0∈L2(Ω,H0

h)such that

(U0,∇χ)=0for all χ∈Lh.

1. For given Um∈L2(Ω; H0

h), ﬁnd

Um+1 ∈L2(Ω; H0

h) such that P-a.s.

Um+1 −Um,Ψ

Ψ

Ψ+k∇

Um+1,∇Ψ

Ψ

Ψ

=kfm+1,Ψ

Ψ

Ψ+B(

Um)ΔWm+1,ψ

ψ

ψ∀Ψ

Ψ

Ψ∈H0

h.(5.1)

2. For given

Um+1 ∈L2(Ω; H0

h(D)), compute Pm+1 ∈L2(Ω; Lh) such that P-a.s.

∇Pm+1,∇χ=1

k

Um+1,∇χ∀χ∈Lh.(5.2)

3. Update P-a.s.

Um+1,ϕ

ϕ

ϕ=

Um+1,ϕ

ϕ

ϕ−k∇Pm+1,ϕ

ϕ

ϕ∀ϕ

ϕ

ϕ∈H0

h.

The following result provides error estimates for the fully discrete scheme.

Theorem 5.1. Suppose that the assumptions in Lemma 3.1 hold. Let {

Um}m≥1⊂

L2(Ω; H0

h)be compu ted from Algor ith m 5.1. Then

max

1≤m≤MEu(tm,·)−

Um2

L21/2

+ Ek

M

m=1 ∇[u(tm,·)−

Um]2

L2!1/2

≤C√k+h+h2

√k.

Because of Theorem 3.1, it is suﬃcient to control the error between the solutions

of Algorithms 1.1 and 5.1, for which Lemma 3.1 is relevant. Balancing the coupling

error O(h2

√k) with the other two errors due to time discretization, splitting, and spatial

discretization motivates a (noncritical) balancing h≤C√k. We remark that this

coupling is well-known in the deterministic setting, where stability of equal-order

ﬁnite element pairings using the pressure stabilization ansatz

div uε−εΔpε=0 inD, ∂npε=0 on∂D

requires choices ε≥Ch2; cf. [12, 16]: since ε=kin (5.2), the restriction k≥Ch2

then leads to a stable discretization in space by equal-order ﬁnite element pairings.

Proof. For every m≥1, let

em,η

m:=

um−

Um,p

m−Pm∈L2Ω; H1

0×H1/R

be the solution of the following set of error equations, which hold P-a.s.:

em+1 −em,Ψ

Ψ

Ψ+k∇em+1,∇Ψ

Ψ

Ψ+k∇ηm,Ψ

Ψ

Ψ

(5.3)

=[B(

um)−B(

Um)]ΔWm+1,Ψ

Ψ

Ψ∀Ψ

Ψ

Ψ∈H0

h,

div em+1,χ

+k∇ηm+1,∇χ=0 ∀χ∈Lh,(5.4)

and e0L2≤Ch2in D.Observethatη0= 0 is a consistent choice, taking into

account (1.7), (1.8), (5.2), together with the fact that (U0,∇χ) = 0 for all χ∈Lh.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

2932 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

The equations follow from the reformulation of Algorithm 1.1 in the form (1.6)–(1.8)

and corresponding equations for (5.1), (5.2). We may choose Ψ

Ψ

Ψ=P0

hem+1 as a test

function in (5.3). We use Young’s inequality and L2-stability of P0

hto conclude for

any δ1>0,

1

2em+12

L2−em2

L2+em+1 −em2

L2+3k

4∇em+12

L2+k∇ηm,P0

hem+1

≤[B(

um)−B(

Um)]ΔWm+1,P0

hem+k∇[

um+1 −P0

h

um+1]2

L2

(5.5)

+Cδ1

[B(

um)−B(

Um)]ΔWm+1

2

L2+δ1em+1 −em2

L2.

A lower bound for the last term on the left-hand side is as follows (δ2>0):

k∇ηm,P0

hem+1≥k∇P1

hηm,em+1−δ2k2∇ηm2

L2−1

4δ2

um+1 −P0

h

um+12

L2

−kpm−P1

hpm2

L2−k

4∇em+12

L2.(5.6)

We use χ=P1

hηmin (5.4) to conclude k(∇P1

hηm,em+1)=k2(∇P1

hηm,∇ηm+1). We

use properties of P1

hto conclude

k2∇P1

hηm,∇ηm+1=k2∇P1

hηm+1,∇ηm+1 −k2∇P1

h[ηm+1 −ηm],∇ηm+1

=k2∇ηm+12+k2∇[pm+1 −P1

hpm+1],∇ηm+1

−k2∇[ηm+1 −ηm],∇P1

hηm+1.

Because of (5.4) we may now conclude (δ3,δ

4>0)

k2∇P1

hηm,∇ηm+1=k21−δ3∇ηm+12

L2−Cδ3k2∇[pm+1 −P1

hpm+1]2

L2

−kem+1 −em,∇P1

hηm+1

≥k21−δ3−δ4∇ηm+12

L2−Cδ3k2∇pm+12

L2

−1

4δ4em+1 −em2

L2.

Because of standard approximation results and Lemma 3.1, arising interpolation

error terms in (5.5)–(5.6) may be controlled as follows:

Ek

M+1

m=1 ∇(

um−P0

h

um)2

L2≤Ch2Ek

M+1

m=1 Δ

um2

L2≤Ch2,

EM+1

m=1

um−P0

h

um2

L2≤Ch4EM+1

m=1 Δ

um2

L2≤Ch4

k,(5.7)

Ek

M+1

m=1 pm−P1

hpm2

L2≤Ch2Ek

M+1

m=1 ∇pm2

L2≤Ch2,

where (5.7)2comes from (5.6), which involves a coupling of discretization scales in

space and time.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

THE STOCHASTIC STOKES EQUATION 2933

To keep the corresponding terms in (5.8) nonnegative, it is possible to choose

δi>0 such that

1−δ2−δ3−δ4>0and 1

2−δ1−1

4δ4≥0.

Next, we sum over all 0 ≤m≤m∗≤M−1 in (3.10), and take expectations. Then,

by the discrete Gronwall inequality and (5.7),

Eem∗+12

L2+1

2−δ1−1

4δ4Em∗

m=0 em+1 −em2

L2+Ek

m∗

m=0 ∇em+12

L2

+k21−δ2−δ3−δ4Em∗

m=0 ∇ηm+12

L2≤Ctm∗h2+h4

k.(5.8)

This proves the theorem.

Remark 5.1. The same techniques may be used to ﬁnd a corresponding error

bound for the ﬁnite element discretization of Algorithm 4.1.

6. Computational experiments. In this section, we report on comparative

computational studies for both the Euler method (1.10)–(1.11) and the splitting Al-

gorithm 5.1. For a stable discretization in space, we use the LBB-stable MINI ele-

ment; cf. [2, 12] for details. In this section we assume that (1.1) is driven by a ﬁnite

dimensional noise. For the underlying domain D=(0,1)2⊂R2and a determinis-

tic applied forcing term f, we consider the constant ﬁnite dimensional forcing term

Bt, u(t)≡B∈L

2(H

H

H,K

K

K), where H

H

H,K

K

K⊂H1are ﬁnite dimensional subsets; see

below. Then

t

0

BdW(s)=

N

j,k=1 t

0

λj,k dβj,k(s)ej,k =

N

j,k=1

λj,k βj,k(t)ej,k (1 ≤N<∞),

where {βj,k}N

j,k=1 are independent R-valued Wiener processes and {ej,k}N

j,k=1 are or-

thonormal functions. Since the above sum is ﬁnite, the operator Bis Hilbert–Schmidt.

The orthonormal functions ej,k are deﬁned by ej,k =gj,kgj,k −1

L2,where

(i) nonsolenoidal functions

gj,k(x, y ):=sin(jπx) sin(kπy),sin(jπx) sin(kπy)and

(ii) solenoidal functions

gj(x, y):=−cos jπx −π

2sin jπy −π

2,sin jπx −π

2cos jπy −π

2

are used. Then λj,k =1

(j+k)2gj,kL2. Note that the index in the solenoidal functions

depends only on j, in order to have orthogonality. Hence H

H

H=K

K

K=span{e1,1,...,

eN,N}⊂H1

0for the basis from (i), and H

H

H=K

K

K=span{e1,...,eN}⊂H∩H2in the

case (ii). Thus, the expansion of the noise in the case (ii) is given by

t

0

BdW(s)=

N

j=1

λjβj(t)ej(1 ≤N<∞)

for λj=1

j2gjL2.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

2934 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

In the experiments below we take N= 4 and address the following topics in

sections 6.1 and 6.2:

(A) How does nonsolenoidal (resp., solenoidal) noise aﬀect strong approximation

properties of Algorithm 5.1? Is Theorem 3.1 sharp with respect to the re-

striction to solenoidal noise? Is improved convergence behavior of iterates of

Algorithm 4.1 for general noise observed computationally?

(B) Chorin’s projection scheme in the deterministic setting is known to exhibit

anisotropic error structures for the pressure, such as boundary layers of mag-

nitude O(√k|log k|); cf., e.g., [17]. What may be concluded accordingly in

the stochastic setting for both tra jectories and expectations of pressure iter-

ates of Algorithms 5.1 and 4.1?

It is evident that if compared to Euler’s method the splitting schemes discussed here

cause reduced computational eﬀort, which in particular pays oﬀ in the present stochas-

tic setting, where a signiﬁcant number of realizations have to be computed to obtain

expectations.

For the experiments below we use T= 1 and compute on cartesian meshes of

size h=1

50 , for a number of realizations Np= 3000, a minimum time discretization

parameter k0=1

4096 and a constant operator B(see beginning of the present section).

To approximate strong errors (1 ≤M∗≤M)

(6.1)

EUM∗

k0−UM∗

ki2

L21/2≈⎛

⎝1

Np

Np

=1 UM∗

k0(ω)−UM∗

ki(ω)2

L2⎞

⎠

1/2

(i≥1),

we use UM∗

k0≈u(tM∗,·) as the (approximate) solution to (1.1) which is computed for

the smallest k01, whereas {UM∗

ki}i≥1are obtained from Algorithm 5.1 for ki=2

ik0

with i=1,2,3,....

6.1. Strong errors for diﬀerent noise. We compare computed velocity iter-

ates of both the Euler scheme (1.10)–(1.11) and Algorithm 5.1 for both solenoidal

and nonsolenoidal noise. The theoretical study in the previous sections needed the

uniform bound

(6.2) Ek

M

m=1 ∇qm2

L2≤C

for pressure iterates of (1.10)–(1.11); this property is shown in Lemma 2.1(iii) in the

case of solenoidal noise and in Lemma 3.1 for pressure iterates of Algorithm 1.1 in

this case as well. The computational results in Figure 1 evidence 1

2as convergence

rate at time T=1fortheL2-error of velocity iterates from Algorithm 5.1, which is in

accordance with Theorem 5.1. Figure 2 reports corresponding results for applied non-

solenoidal noise; we observe a reduction of the convergence rate for velocity iterates

of Algorithm 5.1 by approximately 50%, while Euler iterates still converge optimally

in the L2-norm. To further evidence this loss of accuracy for iterates of the splitting

Algorithm 5.1 in the presence of nonsolenoidal noise, our computations in Figure 6

(left) suggest

Ek

M

m=1 ∇Pm2

L2!1/2

≈C

√k,

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

THE STOCHASTIC STOKES EQUATION 2935

10

1

10

2

10

3

10

4

10

−3

10

−2

10

−1

Number of time−steps

Error of the velocity, divergence−free noise

x

−0.5

10

1

10

2

10

3

10

4

10

−3

10

−2

10

−1

Number of time−steps

Error of the velocity, divergence−free noise

x

−0.5

Fig. 1.Solenoidal noise: Rates of convergence for velocity iterates of Algorithm 5.1 (left) and

corresponding Euler iterates from the space discretization of (1.10)–(1.11) (right), both with respect

to the norm given in (6.1).

10

1

10

2

10

3

10

4

10

−2

10

−1

10

0

Number of time−steps

Error of the velocity, non divergence−free noise

x

−0.3

10

1

10

2

10

3

10

4

10

−3

10

−2

10

−1

Number of time−steps

Error of the velocity, non divergence−free noise

x

−0.5

Fig. 2.Nonsolenoidal noise: Rates of convergence for velocity iterates of Algorithm 5.1 (left)

and corresponding Euler iterates from the space discretization of (1.10)–(1.11) (right), both with

respect to the norm given in (6.1).

which is a bound that we obtain for {qm}M

m=1 instead of Lemma 2.1(iii) for applied

nonsolenoidal noise.

6.2. Approximation of pressures. The reformulation (1.6)–(1.8) of Algo-

rithm 1.1 evidences error eﬀects due to homogeneous boundary conditions, which are

well-known in the deterministic setting to cause artiﬁcial boundary layers of thickness

O(√k|log k|); see, e.g., [10, 16, 17] and the literature cited in these works. Hence, it is

reasonable to ask if corresponding anisotropic errors for pressure iterates from Algo-

rithm 1.1 occur in the stochastic setting as well. We remark that no results regarding

(rates of) convergence of iterates {Pm}m≥1from Algorithm 1.1 were obtained in the

previous sections. The following results show error proﬁles for the pressure computed

by Algorithm 5.1 both pathwise and expectationwise, computed for h=1/30 and

k0=1/512. Again, we distinguish between computations for applied solenoidal and

nonsolenoidal noise.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

2936 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

Fig. 3.Solenoidal noise: Error of pressure from Algorithm 5.1 at T=1for ki=1

512 ,1

256 ,1

128

for one realization (ﬁrst line) and its expectation (second line).

Pressure error functions in the case of solenoidal noise for diﬀerent time-step

sizes are depicted in Figure 3 for both a single path (ﬁrst line) and expectations

(second line); in both cases, we observe an anisotropic structure of error proﬁles,

which for expectations are similar to the corresponding deterministic scenario and

are more pronounced in the case of single paths, which grow for increasing time-steps

ki>0.

The inﬂuence of applied nonsolenoidal noise on the accuracy of pressure iterates

can be deduced from the plots in Figure 4: no local error structures are visible for

a single realization; this is diﬀerent from corresponding plots for expectations which

still show boundary layers that dominate error proﬁles and increase for growing values

ki>0.

6.3. Stochastic pressure correction. Here we give some numerical motiva-

tions for the new Algorithm 4.1 by considering the same setting as at the beginning

of this section. Figure 5 shows error plots for diﬀerent types of noise. We observe an

improvement in the case of general noise to almost optimal order, which is rooted in

the improved regularity of the deterministic pressure, which is exclusively needed to

make sure optimal convergence behavior of this time-splitting scheme,

Ek

M

m=1 ∇pm2

L2≤C.

This is shown in Figure 6 for our example with nonsolenoidal noise. There the function

ki→ Eki

M

m=1 ∇Pm

ki2

L21/2

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

THE STOCHASTIC STOKES EQUATION 2937

Fig. 4.Nonsolenoidal noise: Error of pressure from Algorithm 5.1 at T=1for ki=

1

512 ,1

256 ,1

128 for one realization (ﬁrst line) and its expectation (second line).

101102103104

10−3

10−2

10−1

Number of time−steps

Strong error

x

0.5

101102103104

10−3

10−2

10−1

Number of time−steps

Strong error

x

0.5

Fig. 5.Rates of convergence for velocity iterates of Algorithm 4.1 with nonsolenoidal noise

(left) and solenoidal noise (right), plotted with respect to the number of time-steps, both with respect

to the norm given in (6.1).

is plotted for the Chorin scheme (left) and for the scheme with the stochastic pressure

correction, showing the norm of the pressure for small time-steps. Our result suggests

that the deterministic pressure from Algorithm 4.1 has signiﬁcantly better regularity

properties than the pressure from Algorithm 5.1. We conjecture that the observed

reduced growth with respect to the time-step k>0 for the deterministic pressure

is due to interacting boundary layer eﬀects and space discretization eﬀects of the

nonsolenoidal noise.

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

2938 E. CARELLI, E. HAUSENBLAS, AND A. PROHL

10

1

10

2

10

3

10

4

10

0

10

1

10

2

10

3

Number of time−steps

Pressure

x

0.6

10

1

10

2

10

3

10

4

10

0

10

1

10

2

10

3

Numberof time−steps

Stochastic pressure

x

0.6

10

1

10

2

10

3

10

4

10

0

10

1

10

2

Numberof time−steps

Deterministic pressure

x

0.35

Fig. 6.Nonsolenoidal noise: Evolution of the H1/R-norm of pressure iterates for Algorithm 5.1

(left), stochastic pressure (middle), and deterministic pressure for Algorithm 4.1.

Acknowledgment. The authors gratefully acknowledge interesting discussions

on the subject with Z. Brzezniak (University of York) and S. Peszat (Polish Academy

of Sciences).

REFERENCES

[1] A. Bensoussan,Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), pp. 267–304.