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Time-Splitting Methods to Solve the Stochastic Incompressible Stokes Equation

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Abstract

For the stochastic incompressible time-dependent 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.
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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
utΔu+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
ψC
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)) L2T;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 HnH(n1) 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,L2), suppose that u0L2,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 m0. For given umL2,V)and
umL2,H1
0(D)),
find
um+1 L2,H1
0(D)) such that P-a.s.
(1.2)
um+1 umkΔ
um+1 =kfm+1 +B(tm,
umWm+1 on D.
2. Compute um+1 L2,H)andpm+1 L2,H1(D)/R) such that P-a.s.
um+1
um+1 +kpm+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
umkΔ
um+1 +kpm=kfm+1 +B(tm,
umWm+1 on D,(1.6)
div
um+1 kΔpm+1 =0 onD,(1.7)
npm+1 =0 on∂D,(1.8)
and
u0u0on D. We make the following observations: (i) iterates {
um}m0
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:DTRfrom (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 DRdto be a convex polyhedral domain,
u0VH2(D), and fW2,(0,T;L2(D)), the following optimal estimates are
proved in [16, Theorem 6.1]:
(1.9) max
1mMτmu(tm,·)
umL2+ku(tm,·)
umH1Ck,
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
uC([0,T]; VH2)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 pL(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}mL2(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 wmkΔwm+1 +kqm+1 =kfm+1 +B(tm,wmWm+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 ×DR,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
1mMEu(tm,·)wm2
L21/2
(1.12)
+Ek
1mM∇u(tm,·)wm2
L21/2CTk.
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 DRd,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;t0}, and let {βj(t); t0}j,jN, 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); t0}the cylindrical Wiener process on (Ω,F,F,P), which is defined
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THE STOCHASTIC STOKES EQUATION 2921
as
[0,)t→ W(t)=
j=1
βj(t)ej.
For p1andK
K
Kbeing a Hilbert space, we denote by Mp
F(0,T;K
K
K)(:= Mp
F×
[0,T]; K
K
K)) the space of all F-adapted processes belonging to Lp×[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
ϕM2
F(0,T;L2(H
H
H,K
K
K)) we define the stochas-
tic integral t
0ϕ
ϕ
ϕ(s)dW(s), 0 tT, 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)0tT.
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)=VH2(D), which
is endowed with the norm ·
D(A)=A·
L2. Here, PH:L2Hdenotes the
(Leray) projection operator. For Lipschitz domains DRdand on VH2(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+πH1CfL2
(2.1)
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2922 E. CARELLI, E. HAUSENBLAS, AND A. PROHL
if fL2. 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) u0L2Ω,F0,P;Vand fL2Ω×(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)CTvwK
K
Kt[0,T]v,wH1
0,(2.3)
and for vH1
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 uL2(Ω; C([0,T]; H))M2
F(0,T;V) such that for all t[0,T]andallψ
ψ
ψV
there holds P-a.s.,
u(t)
ψ
ψ+t
0u(s),ψ
ψ
ψds=u0
ψ
ψ+t
0f(s)
ψ
ψds+t
0
B(s, u)dW(s)
ψ
ψ.
The existence of a unique strong solution uL2(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)dst[0,T]
is well-known; see, for instance, [15, Theorem 6.19]. Moreover, standard arguments
yield for u0L2,F0,P;V)thatuL2(Ω; C([0,T]; V)) M2
F(0,T;D(A)), and
Esup
0tTu(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) pL1Ω,F,P;W1,0,T;L2(D)
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THE STOCHASTIC STOKES EQUATION 2923
such that P-a.s.
utΔu+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) pL1Ω,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 uεεΔpε=0 onDT,∂
npε=0 on∂DT,
(iii) div uε+εpε
t=0 onDT,p
ε(0) = p(0) on D,
(iv) div uεεΔpε
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 m0, there exists a solution wm+1 L2Ω; Vsuch that
w0=u0and P-a.s.
(wm+1 wm
ϕ
ϕ)+k(wm+1,ϕ
ϕ
ϕ)
=k(fm+1
ϕ
ϕ)+B(tm,wmWm+1
ϕ
ϕϕ
ϕ
ϕV.(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}m1L2Ω; Vbe a soluti on of (2.12),andlet(2.2),
(2.3),(2.4),and(2.10) be valid. Then
(i) max
1mM
Ewm2
L2+EM
m=1 wmwm12
L2+Ek
M
m=1 ∇wm2
L2
CTEu02
L2+Ek
M
m=1 fm2
L2,
(ii) max
1mM
E∇wm2
L2+EM
m=1 ∇[wmwm1]2
L2+Ek
M
m=1 Awm2
L2
CTE∇u02
L2+Ek
M
m=1 fm2
L2,
(iii) Ek
M
m=1 ∇qm2
L2CTE∇u02
L2+Ek
M
m=1 fm2
L2,
where CTC(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
2ab,a=|a|2−|b|2+|ab|2to obtain
1
2wm+12
L2−wm2
L2+wm+1 wm2
L2+k∇wm+12
L2
=kfm+1,wm+1 +B(tm,wmWm+1,wm+1 wm
(2.13)
+B(tm,wmWm+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,wmWm+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,wmWm+1,wm
(2.14)
+B(tm,wmWm+1,wm+1 wm.
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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 m0, 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
L2CkΔwm+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}mL2Ω; 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 DRd,d=2,3, be such that (2.1) holds, and let
(2.2)–(2.4), (2.10) be valid. Denote by uL2(Ω; C([0,T]; V)) L2(Ω; L2(0,T;H2))
the strong solution of (1.1),and{
um}mL2(Ω; H1
0(D)) solves Algorithm 1.1.There
exists a constant CC(E[u02
H1],D
T)>0such that
(3.1)
max
1mMEu(tm,·)
um2
L21/2+ Ek
M
m=1 ∇u(tm,·)
um2
L2!1/2
Ck.
The proof is split into several steps: first, we study solutions {(vm,r
m)}m
L2(Ω; H1
0(D)) ×L2,H1(D)/R) of an auxiliary problem, where P-a.s.
vm+1 vmkΔvm+1 +krm+1 =kfm+1 +B(tm,vmWm+1 on D,
div vm+1 kΔrm+1 =0 onD,(3.2)
nrm+1 =0 on∂D,
and v0u0on 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}mL2(Ω; V) of (2.12)
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2926 E. CARELLI, E. HAUSENBLAS, AND A. PROHL
and {vm}mL2,H1
0(D)) of (3.2):
max
1mMEwmvm2
L21/2+
E
k
1mM∇wmvm2
L2
1/2
+
E
k2
1mM∇qmrm2
L2
1/2
Ck.(3.3)
Let em:= wmvmL2,H1(D)) and χm:= qmrmL2,H1(D)/R). Taking
the difference of (1.10) and (3.2) then leads to P-a.s.
em+1 emkΔem+1 +kχm+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 e00in 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
4emem12
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
L2Ck1+Eem2
L2.
We now take expectation termwise and sum over all steps 0 mmM1;
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
L2CtmEk2
m
m=0 ∇qm+12
L2CTk.(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
1mM
Evm2
H1+Ek
M
m=1 vm2
H2+Ek
M
m=1 ∇rm2
L2CT.
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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 mM1, by taking expectations and absorbing terms we arrive at
1
2E∇vM2
L2+1
2EM1
m=0 ∇(vm+1 vm)2
L2+1
4Ek
M1
m=0 Δvm+12
L2
1
2E∇v02
L2+CEk
M1
m=0 ∇rm+12
L2+CEk
M1
m=0 fm+12
L2
(3.7)
+kEM1
m=0 B(tm,vm)2
L2(H
H
H,H1)+1
4EM1
m=0 ∇[vm+1 vm]2
L2,
whereweusethefactthatE[M
m=1(B(tm,vmWm+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
umL2Ω,H1
0(D)and ηm:= χmpm
L2Ω,H1(D)/R, which are determined by the following system of equations, which
hold P-a.s.:
ε
ε
εm+1 ε
ε
εmkΔε
ε
εm+1 +kηm=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
ε00,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+k2ηm+1,ηm
[B(tm,vm)B(tm,
um)]ΔWm+1
ε
εmk(rm+1 rm)
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
L2CTk1+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
k2ηm+1,ηm=k2∇ηm+1 2
L2k2ηm+1,[ηm+1 ηm]
=k2∇ηm+12
L2kηm+1
ε
εm+1 ε
ε
εm
(3.11)
11
4δ2k2∇ηm+12
L2δ2ε
ε
εm+1 ε
ε
εm2
L2.
The remaining crucial term in (3.10) is bounded as follows:
k2[rm+1 rm],ηm+1k2δ3∇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
11
4δ2δ30and1
2δ1δ20.
Next, we sum over all 0 mmM1 in (3.10) and take expectations. Then,
by the discrete version of Gronwall’s inequality,
Eε
ε
εm+12
L2+1
2δ1δ2Em
m=0 ε
ε
εm+1 ε
ε
εm2
L2+Ek
m
m=0 ∇ε
ε
εm+12
L2
+k211
4δ2δ3Em
m=0 ∇ηm+12
L2CT4k2
4δ3
m+1
m=0 ∇rm2
L2CtTk,(3.12)
where the last estimate uses (3.6)3and r00 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
1mMEu(tm,·)
um2
L21/2
+
E
k
1mM∇u(tm,·)
um2
L2
1/2
Ck+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,DRd,d=2,3, be such that (2.1) holds, and let (2.2)
(2.4),(2.10) be valid. Let {
um}m1L2,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˜
umL2(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.
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THE STOCHASTIC STOKES EQUATION 2929
We now sum up 0 mMand may use the available bound E[kM
m=1 ∇pm2
L2]
Cto obtain
Ek
M
m=1 Δ
um2
L2CT.
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,
umWm+1}m0approximates general noise. The scheme that we propose
is the following.
Algorithm 4.1. Let m0.
1. For given
umL2,H1
0(D)), find ξ
ξ
ξm+1 L2,H) such that P-a.s.
ξ
ξ
ξm+1 +sm+1 =1
kB(tm,
umWm+1 on D,
div ξ
ξ
ξm+1 =0 onD,(4.1)
ξ
ξ
ξm+1,n=0 on∂D.
2. For given umL2,H), find
um+1 L2,H1
0(D)) such that P-a.s.
(4.2)
um+1 umkΔ
um+1 =kfm+1 +
ξ
ξm+1 on D.
3. Compute um+1 L2,H)andpm+1 L2,H1(D)/R) from the following
equations, which hold P-a.s.:
um+1
um+1 +kpm+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}m1in
(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}m1is involved. This argument is further detailed by the following
formal computation for Euler iterates from (1.10)–(1.11):
wm+1 kΔwm+1 +kpm+1 =wm+kfm+1 +B(tm,wmWm+1
=wm+kfm+1 +
ξ
ξm+1 +ksm+1,
where ξ
ξ
ξm+1 =1
kPH[B(tm,wmWm+1]. As a consequence, we get
wm+1 kΔwm+1 +kπm+1 =wm+kfm+1 +PHB(tm,wmWm+1,
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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 vH1
0(D) the projection PHvH1(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,DRd,d=2,3, be such that (2.1) holds, and let
(2.2)–(2.4) be valid. Denote by uL2(Ω; C([0,T]; Vper)) L2(Ω; L2(0,T;H2
per)) the
strong solution of (1.1),and{
um}mL2,H1
per (D)) solves Algorithm 4.1. There
exists a constant CC(E[u0H1],D
T)>0such that
(4.4)
max
1mMEu(tm,·)
um2
L21/2+ Ek
M
m=1 ∇u(tm,·)
um2
L2!1/2
Ck.
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=(ΦC(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:= HhL2(D)/R
and H0
h:= HhH1
0(D). We recall the L2-orthogonal projection P0
h:L2H0
h,
where
(φ
φ
φP0
hφ
φ
φ, ξ
ξ
ξ=0 ξ
ξ
ξH0
h,
for which holds
φ
φ
φP0
hφ
φ
φL2+h∇(φ
φ
φP0
hφ
φ
φ)L2Ch2φ
φ
φH2φ
φ
φH2.
Accordingly, there holds for P1
h:H1(D)/RLh,where
[χP1
hχ],η=0 ηLh,
that
χP1
hχL2+h∇[χP1
hχ]L2Ch2χH2χH1/RH2.
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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 m0.Set
˜
U0:= U0for U0L2,H0
h)such that
(U0,χ)=0for all χLh.
1. For given UmL2(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(
UmWm+1
ψ
ψΨ
Ψ
ΨH0
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,χχLh.(5.2)
3. Update P-a.s.
Um+1
ϕ
ϕ=
Um+1
ϕ
ϕkPm+1
ϕ
ϕϕ
ϕ
ϕH0
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}m1
L2(Ω; H0
h)be compu ted from Algor ith m 5.1. Then
max
1mMEu(tm,·)
Um2
L21/2
+ Ek
M
m=1 ∇[u(tm,·)
Um]2
L2!1/2
Ck+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 hCk. 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 uεεΔpε=0 inD, ∂npε=0 on∂D
requires choices εCh2; cf. [12, 16]: since ε=kin (5.2), the restriction kCh2
then leads to a stable discretization in space by equal-order finite element pairings.
Proof. For every m1, let
em
m:=
um
Um,p
mPmL2Ω; 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+kem+1,Ψ
Ψ
Ψ+kηm,Ψ
Ψ
Ψ
(5.3)
=[B(
um)B(
Um)]ΔWm+1,Ψ
Ψ
ΨΨ
Ψ
ΨH0
h,
div em+1
+kηm+1,χ=0 χLh,(5.4)
and e0L2Ch2in 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 χLh.
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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+kηm,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):
kηm,P0
hem+1kP1
hηm,em+1δ2k2∇ηm2
L21
4δ2
um+1 P0
h
um+12
L2
kpmP1
hpm2
L2k
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
k2P1
hηm,ηm+1=k2P1
hηm+1,ηm+1 k2P1
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
4>0)
k2P1
hηm,ηm+1=k21δ3∇ηm+12
L2Cδ3k2∇[pm+1 P1
hpm+1]2
L2
kem+1 em,P1
hηm+1
k21δ3δ4∇ηm+12
L2Cδ3k2∇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 ∇(
umP0
h
um)2
L2Ch2Ek
M+1
m=1 Δ
um2
L2Ch2,
EM+1
m=1
umP0
h
um2
L2Ch4EM+1
m=1 Δ
um2
L2Ch4
k,(5.7)
Ek
M+1
m=1 pmP1
hpm2
L2Ch2Ek
M+1
m=1 ∇pm2
L2Ch2,
where (5.7)2comes from (5.6), which involves a coupling of discretization scales in
space and time.
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THE STOCHASTIC STOKES EQUATION 2933
To keep the corresponding terms in (5.8) nonnegative, it is possible to choose
δi>0 such that
1δ2δ3δ4>0and 1
2δ11
4δ40.
Next, we sum over all 0 mmM1 in (3.10), and take expectations. Then,
by the discrete Gronwall inequality and (5.7),
Eem+12
L2+1
2δ11
4δ4Em
m=0 em+1 em2
L2+Ek
m
m=0 ∇em+12
L2
+k21δ2δ3δ4Em
m=0 ∇ηm+12
L2Ctmh2+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)2R2and 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}⊂HH2in 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.
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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 MM)
(6.1)
EUM
k0UM
ki2
L21/2
1
Np
Np
=1 UM
k0(ω)UM
ki(ω)2
L2
1/2
(i1),
we use UM
k0u(tM,·) as the (approximate) solution to (1.1) which is computed for
the smallest k01, whereas {UM
ki}i1are 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
L2C
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,
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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}m1from 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.
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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
L2C.
This is shown in Figure 6 for our example with nonsolenoidal noise. There the function
ki→ Eki
M
m=1 ∇Pm
ki2
L21/2
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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.
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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).
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