Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise

Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise

Rikard Anton Department of Mathematics and Mathematical Statistics, Umeå University, SE–901 87 Umeå, Sweden (rikard.anton@umu.se).    David Cohen Department of Mathematics and Mathematical Statistics, Umeå University, SE–901 87 Umeå, Sweden (david.cohen@umu.se). Department of Mathematics, University of Innsbruck, A–6020 Innsbruck, Austria (david.cohen@uibk.ac.at)    Stig Larsson Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE–412 96 Gothenburg, Sweden (stig@chalmers.se).    Xiaojie Wang School of Mathematics and Statistics, Central South University, CN–410083 Changsha, Hunan, China (x.j.wang7@csu.edu.cn)
Abstract

A fully discrete approximation of the semi-linear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space and a stochastic trigonometric method for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretisation and thus do not suffer from a step size restriction as in the often used Störmer-Verlet-leap-frog scheme. Furthermore, it satisfies an almost trace formula (i. e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.

Key words. Semi-linear stochastic wave equation, Multiplicative noise, Strong convergence, Trace formula, Stochastic trigonometric methods, Geometric numerical integration

AMS subject classifications. 65C20, 60H10, 60H15, 60H35, 65C30

1 Introduction

We consider the numerical discretisation of semi-linear stochastic
wave equations of the form

 d˙u−Δudt=f(u)dt+g(u)dW in D×(0,∞), \hb@xt@.01(1.1) u=0 in ∂D×(0,∞), u(⋅,0)=u0, ˙u(⋅,0)=v0 in D,

where and , , is a bounded convex domain with polygonal boundary . The “” denotes the time derivative . Assumptions on the smoothness of the nonlinearities and will be given below. The stochastic process is an -valued (possibly cylindrical) -Wiener process with respect to a normal filtration on a filtered probability space . The initial data and are -measurable random variables. We will numerically solve this problem with a linear finite element method in space and a stochastic trigonometric method in time.

We refer to the introductions of [kls10] and [cls13] for the relevant literature on the spatial, respectively temporal, discretisation of stochastic (linear) wave equations. Further, the recent publication [wgt13] presents a full discretisation of the wave equation with additive noise: a spectral Galerkin approximation is used in space and an adapted stochastic trigonometric method, using linear functionals of the noise as in [jk09], is employed in time. Furthermore, the time discretisation of nonlinear stochastic wave equations by stochastic trigonometric methods is analysed in [raey]. Finally, let us mention the recent publication [cqs14] which analyses convergence in of the stochastic trigonometric method applied to the one-dimensional nonlinear stochastic wave equation.

In the present publication, we prove mean-square convergence for the full discretisation to the exact solution to the nonlinear problem (LABEL:swe). Furthermore, using this result, we derive a geometric property of our numerical integrator, namely a trace formula. The trace formula (the linear drift of the expected value of the energy) for the exact solution of (LABEL:swe) as well as for the finite element solution and the completely discrete solution are presented.

Strong approximations of stochastic wave equations are relevant in many real applications. For example, let us consider the motion of a strand of DNA floating in a liquid as presented in [MR2508773] and references therein. The motion of the DNA molecule may be modeled by a wave equation and the impact of the fluid’s molecules may be modeled by a stochastic force acting on the string. When two normally distant parts of the DNA get close enough, biological events, such as release of enzymes, occur. It is thus of interest to consider strong approximation of stochastic wave equations in such a situation.

The paper is organised as follows. We introduce some notations and mention some useful results in the next section. Section LABEL:sect:ms presents a mean-square convergence analysis for our numerical discretisation. A trace formula for the exact and numerical solutions is given in Section LABEL:sect:trace. Finally, numerical experiments illustrating the rates of convergence and the trace formula of the numerical solution are given in the final section.

2 Notations and useful results

Let and be separable Hilbert spaces with norms and respectively. We denote the space of bounded linear operators from to by , and we let be the set of Hilbert-Schmidt operators with norm

 ∥T∥L2(U,H):=(∞∑k=1∥Tek∥2H)1/2,

where is an arbitrary orthonormal basis of . If , then we write and . Let be a self-adjoint, positive semidefinite operator. We denote the space of Hilbert-Schmidt operators from to by with norm

 ∥T∥L02=∥TQ1/2∥HS.

For the stochastic wave equation (LABEL:swe), we define and denote the -norm by . Further, we set with .

Let be a filtered probability space and the space of -valued square integrable random variables with norm

 ∥v∥L2(Ω,H):=E[∥v∥2H]1/2.

Next, we define the space , for with norm

 ∥v∥α:=∥Λα/2v∥L2(D)=(∞∑j=1λαj(v,φj)2L2(D))1/2,

where are the eigenpairs of with orthonormal eigenvectors. We also introduce the space

 Hα:=˙Hα×˙Hα−1,

with norm for and . Note that and . In the following we denote the scalar product by and recall the notation for the norm .

Denoting the velocity of the solution to our stochastic partial differential equation by , one can rewrite (LABEL:swe) as

 dX(t)=AX(t)dt+F(X(t))dt+G(X(t))dW(t),  t>0,X(0)=X0, \hb@xt@.01(2.1)

where , , , and . The operator with is the generator of a strongly continuous semigroup of bounded linear operators on , in fact, a unitary group.

Let be a quasi-uniform family of triangulations of the convex polygonal domain with and . Let be the space of piecewise linear continuous functions with respect to which are zero on the boundary of , and let denote the -orthogonal projector and the -orthogonal projector (Ritz projector). Thus,

 (Phv,wh)=(v,wh),(∇Rhu,∇wh)=(∇u,∇wh),∀v∈˙H0,u∈˙H1,wh∈Vh.

The discrete Laplace operator is then defined by

 (Λhvh,wh)=(∇vh,∇wh)∀wh∈Vh.

We note that . We also define discrete variants of and by

 ∥vh∥h,α=∥Λα/2hvh∥,vh∈Vh

and equipped with the norm . Finally, the finite element approximation of (LABEL:swe) can then be written as

 d˙uh,1(t)+Λhuh,1(t)dt=Phf(uh,1(t))dt+Phg(uh,1(t))dW(t),  t>0,uh,1(0)=uh,0, uh,2(0)=vh,0, \hb@xt@.01(2.2)

or in the abstract form

 dXh(t)=AhXh(t)dt+PhF(Xh(t))dt+PhG(Xh(t))dW(t),  t>0,Xh(0)=Xh,0, \hb@xt@.01(2.3)

where , , and are as before, and with . Note the abuse of notation for the projection and similarly for . This will be used throughout the paper. Again, is the generator of a -semigroup on .

We study the equations (LABEL:swe2) and (LABEL:femswe2) in their mild form

 X(t) =E(t)X0+∫t0E(t−s)F(X(s))ds+∫t0E(t−s)G(X(s))dW(s), \hb@xt@.01(2.4) Xh(t) =Eh(t)Xh,0+∫t0Eh(t−s)PhF(Xh(s))ds+∫t0Eh(t−s)PhG(Xh(s))dW(s), \hb@xt@.01(2.5)

where the semigroups can be expressed as

 E(t) =[C(t)Λ−1/2S(t)−Λ1/2S(t)C(t)], \hb@xt@.01(2.6) Eh(t) =⎡⎣Ch(t)Λ−1/2hSh(t)−Λ1/2hSh(t)Ch(t)⎤⎦, \hb@xt@.01(2.7)

with , , and .

In order to ensure existence and uniqueness of problem (LABEL:swe) we shall assume that and , with for some regularity parameter , and that the functions and satisfy

 ∥f(u)−f(v)∥+∥g(u)−g(v)∥L02 ≤C∥u−v∥, if β≥0, \hb@xt@.01(2.8) ≤C(1+∥u∥), if 0≤β≤1, ∥Λ(β−1)/2f(u)∥+∥Λ(β−1)/2g(u)∥L02 ≤C(1+∥Λ(β−1)/2u∥), if β>1,

for all in the first two inequalities and for all in the last one. Through the text, (or etc.) denotes a generic positive constant that may vary from line to line. We assume that the order of initial regularity so that the discrete initial value is well defined.

Lemma 2.1

Assume that , with and the functions and satisfy (LABEL:assFG) for some . Then there exists a unique solution to the stochastic wave equation (LABEL:swe2) and the finite element equation (LABEL:femswe2) given by the solution of their respective mild equation, i. e., equations (LABEL:exactsol) and (LABEL:exactsolfem).

The proof of this lemma follows from [DaPrato1992, Theorem 7.4], see also the proof of Theorem 2.1 in [raey].

We now collect some results that we will use later on. Sketches of the proofs of these results are collected in the appendix at the end of this paper.

The error estimates for the cosine and sine operators (Corollary 4.2 in [kls10]): Denote and let

 Gh(t)X0 =(Ch(t)Rh−C(t))u0+(Λ−1/2hSh(t)Ph−Λ−1/2S(t))v0, ˙Gh(t)X0 =−(Λ1/2hSh(t)Rh−Λ1/2S(t))u0+(Ch(t)Ph−C(t))v0.

Then we have

 ∥Gh(t)X0∥≤C⋅(1+t)⋅hγ−1|||X0|||γ,t≥0,γ∈[1,3],∥˙Gh(t)X0∥≤C⋅(1+t)⋅h23(γ−1)|||X0|||γ,t≥0,γ∈[1,4]. \hb@xt@.01(2.9)

These will be used to estimate the error contributions from the initial values. In order to deal with the convolution terms in (LABEL:exactsolfem) we single out the following error estimates. Let

 Kh(t)v0 =(Λ−1/2hSh(t)Ph−Λ−1/2S(t))v0, ˙Kh(t)v0 =(Ch(t)Ph−C(t))v0.

Then we have

 ∥Kh(t)v0∥≤C⋅(1+t)⋅h23β∥v0∥β−1,t≥0,β∈[0,3],∥˙Kh(t)v0∥≤C⋅(1+t)⋅h23(β−1)∥v0∥β−1,t≥0,β∈[1,4]. \hb@xt@.01(2.10)

The temporal Hölder continuity of the sine and cosine operators, see in [cls13]:

 ∥(Sh(t)−Sh(s))Λ−β/2h∥L(U) ≤C⋅|t−s|β, β∈[0,1], \hb@xt@.01(2.11) ∥(Ch(t)−Ch(s))Λ−(β−1)/2h∥L(U) ≤C⋅|t−s|β−1, β∈[1,2],

together with its continuous version:

 ∥(S(t)−S(s))Λ−β/2∥L(U) ≤C⋅|t−s|β, β∈[0,1], \hb@xt@.01(2.12) ∥(C(t)−C(s))Λ−(β−1)/2∥L(U) ≤C⋅|t−s|β−1, β∈[1,2].

The equivalence of and , see the proof of Theorem 4.4 in [kll]: This uses an inverse inequality, hence our assumption about the quasi-uniformity of the mesh family.

 ∥ΛαhPhΛ−αv∥2≤∥v∥2,  α∈[−12,1],  v∈˙H0=L2(D). \hb@xt@.01(2.13)

The equivalence of the discrete and continuous norm, see in [al13]:

 c∥Λγhvh∥≤∥Λγvh∥≤C∥Λγhvh∥forvh∈Vhandγ∈[−12,12]. \hb@xt@.01(2.14)

Using the above estimates, one can deduce the following regularity results for the exact solution to our stochastic wave equation (LABEL:swe) and for the exact solution of the finite element approximation (LABEL:femswe1).

Proposition 2.2

Let be the solution to (LABEL:swe), where the initial values satisfy , with , and the functions and satisfy (LABEL:assFG) for some . Then it holds that

 sup0≤t≤TE[∥u1(t)∥2β+∥u2(t)∥2β−1]≤C

and, for ,

 E[∥u1(t)−u1(s)∥2] ≤C|t−s|2min(β,1)(E[∥u0∥2β+∥v0∥2β−1] +supr∈[0,T]E[1+∥u1(r)∥2β]).

The proof of this proposition is very similar to the proof of Proposition LABEL:prop:regFEM given below and is therefore omitted (see also the proofs of Proposition 3.1 and Lemma 3.3 in [raey]).

The next result will be useful in Section LABEL:sect:trace when we will deal with the trace formula of the numerical solution.

Proposition 2.3

Let be the solution to the finite element problem (LABEL:femswe1), where the initial values satisfy , with , and the functions and satisfy (LABEL:assFG) for some . Then it holds that

 sup0≤t≤TE[∥uh,1(t)∥2h,β+∥uh,2(t)∥2h,β−1]≤C

and for

 E[∥uh,1(t)−uh,1(s)∥2] ≤C|t−s|2min(β,1)(E[∥uh,0∥2h,β+∥vh,0∥2h,β−1] +supr∈[0,T]E[1+∥uh,1(r)∥2h,β]),

where we recall that and are the initial position and velocity to the finite element problem.

Proof. Let us start with the first estimate of the norm of and consider the expression

 Λβ/2huh,1(t) =Λβ/2hCh(t)uh,0+Λ(β−1)/2hSh(t)vh,0 +∫t0Λ(β−1)/2hSh(t−r)Phf(uh,1(r))dr +∫t0Λ(β−1)/2hSh(t−r)Phg(uh,1(r))dW(r).

Using the fact that and commute, the boundedness of the cosine operator, together with our assumptions on the initial values for the finite element problem, we get

 E[∥Λβ/2hCh(t)uh,0∥2]≤Cforβ∈[0,2].

Similarly, one obtains

 E[∥Λ(β−1)/2hSh(t)vh,0∥2]≤C.

To estimate the third term, we use (LABEL:lpl), the assumptions on given in (LABEL:assFG), and the equivalence of the norms stated in (LABEL:equivnorm). First for , we get

 E[∥∥∫t0Λ(β−1)/2hSh(t−r)Phf(uh,1(r))dr∥∥2] ≤C1+C2∫t0E[∥uh,1(r)∥2]dr ≤C3+C4∫t0E[∥uh,1(r)∥2h,β]dr,

because and are bounded. For , we have by (LABEL:lpl)

 E[∥∥∫t0Sh(t−r)Λ(β−1)/2hPhΛ−(β−1)/2Λ(β−1)/2f(uh,1(r))dr∥∥2] ≤C∫t0E[1+∥Λ(β−1)/2uh,1(r)∥2]dr≤C1+C2∫t0E[∥uh,1(r)∥2h,β−1]dr ≤C3+C4∫t0E[∥uh,1(r)∥2h,β]dr.

Finally, Ito’s isometry, equations (LABEL:equivnorm) and (LABEL:lpl), and the assumptions (LABEL:assFG) on give us

All together, for , one thus obtains

 E[∥uh,1(t)∥2h,β]≤K1+K2∫t0E[∥uh,1(r)∥2h,β]dr

and an application of Gronwall’s lemma give the desired bound for .

The proof for the other bound is done in the same way except for a slight difference in the initial values and that in the integrals is replaced by .

We now prove a Hölder regularity property of the finite element solution. We write, for ,

 uh,1(t)−uh,1(s) =(Ch(t)−Ch(s))uh,0+Λ−1/2h(Sh(t)−Sh(s))vh,0 +∫s0Λ−1/2h(Sh(t−r)−Sh(s−r))Phf(uh,1(r))dr +∫tsΛ−1/2hSh(t−r)Phf(uh,1(r))dr +∫s0Λ−1/2h(Sh(t−r)−Sh(s−r))Phg(uh,1(r))dW(r) +∫tsΛ−1/2hSh(t−r)Phg(uh,1(r))dW(r).

To estimate the first term we use (LABEL:o41cls) to get

 E[∥(Ch(t)−Ch(s))uh,0∥2] =E[∥(Ch(t)−Ch(s))Λ−β/2hΛβ/2huh,0∥2] ≤C|t−s|2βE[∥Λβ/2huh,0∥2],

for . For we note that and that is bounded in the operator norm. Using a similar argument for the second term, we get the following estimate for the first two terms

 E[∥(Ch(t)−Ch(s))uh,0+Λ−1/2h(Sh(t)−Sh(s))vh,0∥2] ≤C|t−s|2min(β,1)E[∥uh,0∥2h,β+∥vh,0∥2h,β−1],

for . In order to estimate the third term, we use (LABEL:o41cls), the assumptions on , and the equivalence of the norms given in (LABEL:equivnorm). First for , we obtain

 ≤C|t−s|2supt∈[0,T]E[1+∥uh,1(t)∥2] ≤C|t−s|2supt∈[0,T]E[1+∥uh,1(t)∥2h,β].

For we have, using (LABEL:o41cls), (LABEL:lpl), (LABEL:equivnorm) and the fact that is bounded in the operator norm

 ≤∫s0E[∥(Sh(t−r)−Sh(s−r))Λ−1/2hΛ−(β−1)/2hΛ(β−1)/2hPhΛ−(β−1)/2 ×Λ(β−1)/2f(uh,1(r))∥2]dr ≤C|t−s|2supt∈[0,T]E[1+∥uh,1(t)∥2h,β−1] ≤C|t−s|2supt∈[0,T]E[1+∥uh,1(t)∥2h,β].

Similarly we get for the fourth term

 E[∥∥∫tsΛ−1/2hSh(t−r)Phf(uh,1(r))dr∥∥2] ≤C|t−s|2min(β,1)supt∈[0,T]E[1+∥uh,1(t)∥2h,β].

To estimate terms five and six we use Ito’s isometry, (LABEL:o41cls), (LABEL:lpl), (LABEL:equivnorm) and the assumptions on to get, for ,

 E[∥∥∫s0Λ−1/2h(Sh(t−r)−Sh(s−r))Phg(uh,1(r))dW(r)∥∥2] ≤∫s0E[∥(Sh(t−r)−Sh(s−r))Λ−β/2hΛ(β−1)/2hPh ×g(uh,1(r))∥2L02]dr ≤C|t−s|2βsupt∈[0,T]E[∥Λ(β−1)/2hg(uh,1(t))∥2L02] ≤C|t−s|2βsupt∈[0,T]E[1+∥uh,1(t)∥2h,β]

and

 E[∥∥∫tsΛ−1/2hSh(t−r)Phg(uh,1(r))dW(r)∥∥2] ≤∫tsE[∥Sh(t−r)Λ−β/2hΛ(β−1)/2hPhg(uh,1(r))∥2L02]dr ≤C|t−s|2βsupt∈[0,T]E[∥Λ(β−1)/2hg(uh,1(t))∥2L02] ≤C|t−s|2βsupt∈[0,T]E[1+∥uh,1(t)∥2h,β].

For we again use that is bounded in the operator norm.

Collecting the above estimates give us the statement about the regularity of the finite element solution.

3 Mean-square convergence analysis

Recall that the exact solutions to (LABEL:swe2) and (LABEL:femswe2) solve the following equations

 X(t) =E(