FULLY DISCRETE FINITE ELEMENT METHODS FOR NONLINEAR STOCHASTIC ELASTIC WA VE EQUATIONS WITH MULTIPLICATIVE NOISE XIAOBING FENG1 YUKUN LI2 YUJIAN LIN3

2025-05-06 0 0 587.66KB 37 页 10玖币

侵权投诉

FULLY DISCRETE FINITE ELEMENT METHODS FOR NONLINEAR

STOCHASTIC ELASTIC WAVE EQUATIONS WITH MULTIPLICATIVE NOISE

XIAOBING FENG1,∗, YUKUN LI2, YUJIAN LIN3

1Department of Mathematics, The University of Tennessee, Knoxville, TN 37996

2Department of Mathematics, University of Central Florida, Orlando, FL 32816

3Department of Mathematics, Northwestern Polytechnical University, Xian, Shaanxi, China, 710129

Abstract. This paper is concerned with fully discrete ﬁnite element methods for approximating variational solu-

tions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties

of the weak solution is carried out and a fully discrete ﬁnite element method is proposed. Strong convergence in

the energy norm with rate O(k+hr)is proved, where kand hdenote respectively the temporal and spatial mesh

sizes, and r(≥1)is the order of the ﬁnite element. Numerical experiments are provided to test the efﬁciency of

proposed numerical methods and to validate the theoretical error estimate results.

Keywords. Stochastic elastic wave equations; multiplicative noise; Itˆ

o stochastic integral; ﬁnite element method;

error estimates; quantities of stochastic interests.

1. INTRODUCTION

This paper is concerned with numerical approximations of the following stochastic elastic

wave equations with multiplicative noise of Itˆ

o type:

utt −div (σ(u)) = F[u] + G[u]ξin DT:= (0,T)×D,(1.1)

u(0,·) = u0,ut(0,·) = v0in D,(1.2)

u(t,·) = 0 on ∂DT:= (0,T)×∂D,(1.3)

where ut=du

dt,ξ=˙

W=dW

dtis the white noise, D⊂Rd(d=2,3)is a bounded domain, (u0,v0)

is an H1

0×L2-valued random variable, and

σ(u) = (λdiv u)I+µε(u),(1.4)

ε(u) = 1

2∇u+ (∇u)T,(1.5)

F[u] = F(u,∇u),(1.6)

G[u] = G(u,∇u).(1.7)

Here Idenotes the unit matrix. F[u]and G[u]are two given nonlinear mappings satisfying some

structure conditions. The multiplicative noise G[u]ξhas the following three cases:

Case 1. W is a R-valued Wiener process which is deﬁned on the ﬁltered probability space

(Ω,F,{Ft}0≤t≤T,P), and G[u]is d-dimensional nonlinear mapping;

Case 2. Wis a Rd-valued Wiener process , and G[u]is a scalar nonlinear mapping;

Case 3. Wis a Rl-valued Wiener process, G[u]is a d×lmatrix, then G˙

Wis a d-dimensional

multiplicative noise.

arXiv:2210.00592v1 [math.NA] 2 Oct 2022

2 XIAOBING FENG, YUKUN LI, YUJIAN LIN

For the sake of presentation clarity, we only consider Case 1, For the other two cases, it can be

shown that the same results still hold.

Wave propagation is a fundamental physical phenomenon, and it arises from various appli-

cations in geophysics, engineering, medical science, biology, etc. There is a large amount of

literature on numerical methods for deterministic acoustic wave equations, we refer the reader

to [1,3,4,5,7,8,16,17,21,22,29,32,33,35,38,40] and the references therein for a detailed

account. Moreover, numerical methods for stochastic acoustic wave equations have also been

intensively developed in the last few years, see [2,10,12,14,15,18,20,23,24,26,27,30,37].

Similarly, the elastic wave equations are also of great importance and ﬁnd applications in geo-

science for modeling seismic waves and in medical science for tumor detection as well as in

materials science for non-destructive testing. Although there is a large literature in numeri-

cal methods for deterministic elastic wave equations, see [19,25,34,36,28,9,31] and the

references therein, there is barely any work on numerical analysis of stochastic elastic wave

equations in the literature, which motivates us to carry out the work of this paper.

The primary goal of this paper is to develop some semi-discrete (in time) scheme and fully

discrete ﬁnite element methods for the stochastic elastic wave equation with multiplicative

noise. The highlight of the paper is the establishment of strong norm convergence and error

estimates for both semi-discrete and fully discrete methods. To achieve this goal, we ﬁrst need

to establish some stability and H¨

older continuity estimates for the (variational) weak solution

of the stochastic wave equations. These results will be crucially used to derive the desired error

estimates for the semi-discrete scheme. We next need to establish various (energy) stability

estimates for the semi-discrete numerical solution, which are necessary for deriving the desired

error estimates for the fully discrete ﬁnite element methods.

The rest of the paper is organized as follows. In Section 2, we introduce a variational weak

formulation for problem (1.1)–(1.3). The stability and H¨

older continuity estimates in the L2-,

H1-, and H2-norm are established for the strong solution. In Section 3, we propose a semi-

discrete in time numerical scheme for problem(1.1)–(1.3). It is proved that the semi-discrete

solution is energy stable. Moreover, we prove the convergence with rate O(k)in the L2-norm

and O(k1

2)in the H1-norm for the displacement approximations. In Section 4, we propose a

fully discrete ﬁnite element method to discretize the semi-discrete scheme in space and derive its

error estimates, which show that for the linear ﬁnite element, the L2-norm of the error converges

with O(h2)rate and the H1-norm converges with O(h)rate. In Section 5, we present two two-

dimensional numerical experiments to test the efﬁciency of the proposed numerical methods

and to validate the theoretical error estimate results. Finally, we conclude the paper with a short

summary given in Section 6.

2. PRELIMINARIES

Standard notations for functions and spaces are adopted in this paper. For example, Lpde-

notes (Lp(D))dfor 1 ≤p≤∞,(·,·)denotes the standard L2(D)-inner product and Hm(D)

denotes the Sobolev space of order m. Throughout this paper, Cwill be used to denote a generic

positive constant which is independent the mesh parameters kand h.

FINITE ELEMENTS FOR NONLINEAR STOCHASTIC ELASTIC WAVE EQUATIONS 3

2.1. Assumptions. The following structural conditions will be imposed on the mappings F[·]

and G[·]:

kF[0]kL2+kG[0]kL2≤CA,(2.1)

k∇uF[·]kL∞+k∇uG[·]kL∞≤CA,(2.2)

kFuiuj[·]kL∞≤CA,1≤i,j≤d,(2.3)

kF[v]−F[w]kL2≤CBλkdiv(v−w)k2

L2+µkε(v−w)k2

L2+kv−wk2

L21

2,(2.4)

kG[v]−G[w]kL2≤CBλkdiv(v−w)k2

L2+µkε(v−w)k2

L2+kv−wk2

L21

2,(2.5)

where Fuiuj[·]denotes the second derivative of Fwith respect to ui,uj, and CAand CBare two

positive constants.

2.2. Variational weak formulation and properties of weak solutions. In this subsection, we

ﬁrst give the deﬁnition of variational weak formulation and weak solutions for problem (1.1)–

(1.3). We then establish several technical lemmas that will be used in the subsequent sections.

Equations (1.1)–(1.3) can be written as

du=vdt,(2.6)

dv= (Lu+F[u])dt+G[u]dW(t),Lu:=div σ(u),(2.7)

u(0) = u0,v(0) = v0,(2.8)

u(t,·) = 0.(2.9)

Deﬁnition 2.1. The weak formulation for problem (2.6)–(2.9) is deﬁned as seeking (u,v)∈

L2Ω;C([0,T];L2)∩L2((0,T),H1

0)×L2Ω;C([0,T],L2)such that

u(t),φ=Zt

0v(s),φds+ (u0,φ)∀φ∈L2,(2.10)

v(t),ψ=−Zt

λdiv (u(s)),div (ψ)ds−Zt

µε(u(s)),ε(ψ)ds(2.11)

+Zt

0F[u(s)],ψds+Zt

0G[u(s)]dW(s),ψ+ (v0,ψ)∀ψ∈H1

for all (φ,ψ)∈L2×H1

0. Such a pair (u,v), if it exists, is called a (variational) weak solution

to problem (2.6)–(2.9). Moreover, if a weak solution (u,v)belongs to L2Ω;C([0,T];H2∩

0)×L2Ω;C([0,T];H1

0), then (u,v)is called a strong solution to problem (2.6)–(2.9).

Remark 2.2. The well-posedness of problem (2.6)–(2.9) can be proved using the same tech-

nique (i.e., the Galerkin method) as done in [6] for the acoustic stochastic wave equation with

multiplicative noise. The only markable difference is that to verify the coercivity (or ellipticity)

in H1(D)of the operator L, we need to use the well-known Korn’s (second) inequality.

We now state and prove the stability properties of the strong solution (u,v)of problem (2.6)–

(2.9). Those bounds will be used to prove H¨

older continuity in time in this section. They are

also useful in establishing rates of convergence for the numerical schemes.

4 XIAOBING FENG, YUKUN LI, YUJIAN LIN

Lemma 2.3. Let (u,v)be a strong solution to equations (2.6)–(2.9). Under the assumptions

(2.1)–(2.5), there hold

sup

0≤t≤T

Ehkvk2

L2i+sup

0≤t≤T

Ehλkdiv (u)k2

L2+µkε(u)k2

L2i≤Cs1,(2.12)

sup

0≤t≤T

Ehλkdiv (v)k2

L2+µkε(v)k2

L2i+sup

0≤t≤T

EhkLuk2

L2i≤Cs2,(2.13)

sup

0≤t≤T

Eh

∂xjv

2

L2i+sup

0≤t≤T

Ehλ

div (∂xju)

2

L2+µ

ε(∂xju)

2

L2i≤Cs3(2.14)

for 1≤j≤d, and

Cs1=Ehkv0k2

L2i+Ehλkdiv (u0)k2

L2+µkε(u0)k2

L2i+4C2

AeCC2

Cs2=Ehλkdiv (v0)k2

L2+µkε(v0)k2

L2i+EhkLu0k2

L2i+CC2

ACs1eT,

Cs3=Eh

∂xjv0

2

L2i+Ehλ

div (∂xju0)

2

L2+µ

ε(∂xju0)

2

L2ieCC2

Proof. Step 1: Applying Itˆ

o’s formula to the functional Φ1(v(·)) = kv(·)k2

L2yields

kv(t)k2

L2=kv0k2

L2+Zt

DΦ1v(s)Lu+F[u]ds(2.15)

+Zt

TrD2Φ1v(s)G[u],G[u]ds+Zt

DΦ1v(s)G[u]dW(s).

The expressions of DΦ1(v(·)) and D2Φ1(v(·)) are as follows:

DΦ1(v)(w1) = 2(v,w1)∀w1∈C∞

0,(2.16)

D2Φ1(v)(w1,w2) = 2(w1,w2)∀w1,w2∈C∞

Substituting the expressions of DΦ1(v(·)) and D2Φ1(v(·)) into (2.15) , we get

kv(t)k2

L2+λ

div u(t)

2

L2+µ

εu(t)

2

L2(2.17)

=kv0k2

L2+λkdiv (u0)k2

L2+µkε(u0)k2

+2Zt

0F[u],vds+Zt

kG[u]k2

L2ds+2Zt

0G[u],vdW(s)

:=kv0k2

L2+λkdiv (u0)k2

L2+µkε(u0)k2

L2+I1+I2+I3.

For the ﬁrst term I1, let ¯

u=0 in equation (2.4). Using equation (2.1), the Poincar´

e inequality,

and the Korn’s inequality, we have

2Zt

0F[u],vds≤Zt

kF[u]k2

L2+kvk2

L2ds(2.18)

≤2C2

BZt

λkdiv (u)k2

L2+µkε(u)k2

L2+kuk2

L2ds+2C2

A+Zt

0kvk2

L2ds

≤CC2

BZt

λkdiv (u)k2

L2+µkε(u)k2

L2ds+2C2

A+Zt

kvk2

L2ds.

FINITE ELEMENTS FOR NONLINEAR STOCHASTIC ELASTIC WAVE EQUATIONS 5

Taking the expectation on both sides of (2.18), we obtain

2EhZt

0F[u],vdsi≤CC2

BEhZt

λkdiv (u)k2

L2+µkε(u)k2

L2dsi(2.19)

+2C2

A+EhZt

kvk2

L2dsi.

Similarly, using equations (2.1) and (2.5), the Poincar´

e inequality, and the Korn’s inequality,

the expectation of the second term I2can be bounded by

EhZt

kG[u]k2

L2dsi≤2C2

BEhZt

λkdiv (u)k2

L2+µkε(u)k2

L2+kuk2

L2dsi+2C2

A(2.20)

≤CC2

BEhZt

λkdiv (u)k2

L2+µkε(u)k2

L2dsi+2C2

The third term I3is a martingale, and E[I3] = 0. Taking the expectation on both sides of

(2.17) and using the Gronwall’s inequality, we get

Ehkvk2

L2i+Ehλkdiv (u)k2

L2+µkε(u)k2

L2i(2.21)

≤Ehkv0k2

L2i+Ehλkdiv (u0)k2

L2+µkε(u0)k2

L2i+4C2

AeCC2

Step 2: Again, by applying Itˆ

o’s formula to Φ2u(·)=kLu(·)k2

L2, we obtain

kLu(t)k2

L2=kLu0k2

L2+2Zt

0Lu(s),Lv(s)ds.(2.22)

Applying Itˆ

o’s formula to Φ3v(·)=λ

div v(·)

2

L2+µ

εv(·)

2

L2leads to

λkdiv (v(t))k2

L2+µkε(v(t))k2

L2=λkdiv (v0)k2

L2+µkε(v0)k2

L2(2.23)

+2Zt

0λdiv (Lu),div (v)+µε(Lu),ε(v)ds

+2Zt

0λdiv (F[u]),div (v)+µε(F[u]),ε(v)ds

+Zt

0λkdiv (G[u])k2

L2+µkε(G[u])k2

L2ds

+2Zt

0λdiv (G[u]dW(s)),div (v)+µε(G[u]dW(s)),ε(v)ds.

Adding (2.22) and (2.23) gives

kLu(t)k2

L2+λkdiv (v(t))k2

L2+µkε(v(t))k2

L2=λkdiv (v0)k2

L2+µkε(v0)k2

L2(2.24)

+kLu0k2

L2+2Zt

0λdiv (F[u]),div (v)+µε(F[u]),ε(v)ds

+Zt

0λkdiv (G[u])k2

L2+µkε(G[u])k2

L2ds

+2Zt

0λdiv (G[u]dW(s)),div (v)+µε(G[u]dW(s)),ε(v)ds

:=λkdiv (v0)k2

L2+µkε(v0)k2

L2+kLu0k2

L2+I1+I2+I3.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

FULLYDISCRETEFINITEELEMENTMETHODSFORNONLINEARSTOCHASTICELASTICWAVEEQUATIONSWITHMULTIPLICATIVENOISEXIAOBINGFENG1;,YUKUNLI2,YUJIANLIN31DepartmentofMathematics,TheUniversityofTennessee,Knoxville,TN379962DepartmentofMathematics,UniversityofCentralFlorida,Orlando,FL328163DepartmentofMathematics,Northwes...

展开>> 收起<<

FULLY DISCRETE FINITE ELEMENT METHODS FOR NONLINEAR STOCHASTIC ELASTIC WA VE EQUATIONS WITH MULTIPLICATIVE NOISE XIAOBING FENG1 YUKUN LI2 YUJIAN LIN3.pdf

共37页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

FULLY DISCRETE FINITE ELEMENT METHODS FOR NONLINEAR STOCHASTIC ELASTIC WA VE EQUATIONS WITH MULTIPLICATIVE NOISE XIAOBING FENG1 YUKUN LI2 YUJIAN LIN3

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: