# An Approximation of Semigroups Method for Stochastic Parabolic Equations

Authors: Allaberen Ashyralyev; Mehmet Emin San
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A single-step difference scheme for the numerical solution of the nonlocal-boundary
value problem for stochastic parabolic equations is presented. The convergence estimate for the solution
of the difference scheme is established. In application, the convergence estimates for the solution of the
difference scheme are obtained for two nonlocal-boundary value problems. The theoretical statements
for the solution of this difference scheme are supported by numerical examples.


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## Body

## 1.  Introduction

It is known that most problems in heat flow, fusion process, model financial instruments like options, bonds, and interest rates, and other areas which are involved with uncertainty lead to stochastic differential equation with parabolic type. These equations can be derived as models of indeterministic systems and considered as methods for solving boundary value problems.The method of operators as a tool for investigation of the solution to stochastic partial differential equations in Hilbert and Banach spaces has been systematically developed by several authors (see [1–4] and the references therein). Finite difference method for the solution of initial boundary value problem for stochastic differential equations has been studied extensively by many researchers (see [5–15] and the references therein). However, multipoint nonlocal-boundary value problems were not well investigated.In the present paper the multipoint nonlocal-boundary value problem(1.1)dv(t)=-Av(t)dt+f(t)dwt,0<t<Tv(0)=∑j=1J‍αjv(λj)+φ(wλ1,…,wλJ),∑j=1J‍|αj|≤1,0<λ1<⋯<λJ≤T,0≤t≤T
for stochastic parabolic differential equations in a Hilbert space H with a self-adjoint positive definite operator A is considered. Here(i)
wt is a standard Wiener process given on the probability space (Ω,F,P).(ii)
For anyz∈[0,T], f(z) is an element of space Mw2([0,T],H1), where H1 is subspace of H.(iii)
φ(wλ1,…,wλJ) is element of space Mw2([0,T],H2) of H2-valued measurable processes, where H2 is a subspace of H.Here,Mw2([0,T],H)[20] denote the space of H-valued measurable processes which satisfy(a)
ϕ(t) is Ft measurable, a.e. in t,(b)
E∫0T‍∥ϕ(t)∥Hdt<∞.The main goal of this study is to construct and investigate the difference schemes for the multipoint nonlocal-boundary value problems (1.1). The outline of the paper is as follows. In Section 2, the exact single-step difference scheme for the solution of the problem (1.1) is presented. In Section 3, the 1/2-th order of accuracy Rothe difference scheme is constructed and investigated for the approximate solution of the problem (1.1). The estimate of convergence for the solution of this difference scheme is obtained. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of two multipoint nonlocal-boundary value problems for stochastic parabolic equations are obtained. In Section 4, the numerical application for one-dimensional stochastic parabolic equation is presented.


## 2. The Exact Single-Step Difference Scheme

Now, let us give some lemmas we need in the sequel. Throughout this paper, letH be a Hilbert space, let  A be a positive definite self-adjoint operator with A≥δI, where δ>0.Lemma 2.1.
The following estimate holds:(2.1)∥e-tA∥H→H≤e-δt(t≥0),∥Ae-tA∥H→H≤1t(t>0).Lemma 2.2.
Suppose that assumption(2.2)∑k=1J‍|αk|≤1
holds. Then, the operator
(2.3)I-∑k=1J‍αke-λkA
has an inverse
(2.4)Υ=(I-∑k=1J‍αke-λkA)-1,
and the following estimate is satisfied:
(2.5)∥Υ∥H→H≤11-e-λ1δ≤C(δ,λ1).Proof.
The proof follows from the triangle inequality, assumption (2.2), and estimate
(2.6)∥(I-∑k=1J‍αke-λkA)-1∥H→H≤supδ≤μ<∞1|1-∑k=1Jαke-λkμ|.
Let us now obtain the formula for the mild solution of problem (1.1). It is clear that under the assumptions (i)-(ii) and
(2.7)E∥v(0)∥H22<∞,H2⊂H,
the Cauchy problem
(2.8)dv(t)=-Av(t)dt+f(t)dwt,0<t<T,v(0)isgiven
and has a unique mild solution, which is represented by the following formula:
(2.9)v(t)=e-Atv(0)+∫0te-A(t-s)f(s)dws.
Then from this formula and the multipoint nonlocal-boundary condition(2.10)v(0)=∑j=1J‍αjv(λj)+φ(wλ1,…,wλJ),
we get
(2.11)v(0)=∑j=1J‍αje-Aλjv(0)+∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ).
By Lemma 2.2 the operator I-∑j=1Jαje-Aλj has a bounded inverse Υ=(I-∑j=1Jαje-Aλj)-1. Then
(2.12)v(0)=Υ{∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)}.
Therefore, we have formulas (2.9) and (2.12) for the solution of problem (1.1).Now, we will consider the single-step exact difference scheme. On the segment[0,T] we consider the uniform grid space
(2.13)[0,T]τ={tk=kτ,k=0,1,…,N,Nτ=T}
with step τ>0. Here N is a fixed positive integer.Theorem 2.3.
Letv(tk) be the solution of (1.1) at the grid points t=tk. Then {v(tk)}0N is the solution of the multipoint nonlocal-boundary value problem for the following difference equation (see [16]):
(2.14)v(tk)-v(tk-1)+(I-e-τA)v(tk-1)=∫tk-1tk‍e-(tk-s)Af(s)dws,1≤k≤N,(2.15)v(0)=Υ{∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)}.Proof.
Puttingt=tk and t=tk-1 into the formula (2.9), we can write
(2.16)v(tk)=e-tkAv(0)+∫0tke-(tk-s)Af(s)dws,v(tk-1)=e-tk-1Av(0)+∫0tk-1‍e-(tk-1-s)Af(s)dws.
Hence, we obtain the following relation between v(tk) and v(tk-1):
(2.17)v(tk)=e-τAv(tk-1)+∫tk-1tke-(tk-s)Af(s)dws.
Last relation and equality (2.14) are equivalent. Theorem 2.3 is proved.Note that problem (2.14) is called the single-step exact difference scheme for the solution of the problem (1.1).


## 3.   Rothe Difference Scheme

In this section, the1/2-th order of accuracy Rothe difference scheme is constructed and investigated for the approximate solution of the problem (1.1). The estimate of convergence for the solution of this difference scheme is established. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of two multipoint nonlocal-boundary value problems for stochastic parabolic equations are obtained.


### 3.1. 1/2-th Order-of-Accuracy Rothe Difference Scheme

Let us give some lemmas we need in the sequel.Lemma 3.1.
The following estimates hold:(3.1)∥AαRk∥H→H≤1(kτ)α,1≤k≤N,0≤α≤1,(3.2)∥A-α(Rk-e-kτA)∥H→H≤2ταk1-α,1≤k≤N,0≤α≤2,
where R=(I+τA)-1.Lemma 3.2.
Suppose that assumption (2.2) holds. Then, the operator
(3.3)I-∑j=1J‍αjR[λj/τ]
has a bounded inverse
(3.4)Υτ=(I-∑j=1J‍αjR[λj/τ])-1,
and the following estimate is satisfied:
(3.5)∥Υτ∥H→H≤C(δ,λ1).Proof.
The proof follows from the triangle inequality, assumption (2.2), and estimate
(3.6)∥(I-∑j=1J‍αjR[λj/τ])-1∥H→H≤supδ≤μ<∞1|1-∑j=1Jαk(1/(1+μτ)[λk/τ])|.
From (2.14) it is clear that for the approximate solution of the multipoint nonlocal-boundary value problem (1.1) it is necessary to approximate the expressions
(3.7)e-τA,1τ∫tk-1tke-(tk-s)Af(s)dw,
and multipoint nonlocal-boundary condition
(3.8)v(0)=∑j=1J‍αjv(λj)+φ(wλ1,…,wλJ).
It is possible under stronger assumption than (ii) forf(t). Assume that
(3.9)maxt∈[0,T]∥A-1/2f′(t)∥H+maxt∈[0,T]∥A1/2f(t)∥H≤C.
Replacing the expressions e-τA,e-(tk-s)A with R=(I+τA)-1, the expression v(λj) with v([λj/τ]τ), and the function f(s) with f(tk-1), we get the implicit Rothe difference scheme:
(3.10)uk-uk-1+τAuk=f(tk-1)(wtk-wtk-1),1≤k≤N,u0=∑j=1J‍αju[λj/τ]+φ(wλ1,…,wλJ).
Let us now obtain the formula for the solution of problem (3.10). It is clear that the Rothe difference scheme
(3.11)uk-uk-1+τAuk=f(tk-1)(wtk-wtk-1),1≤k≤N,u0isgiven,
for the solution of the Cauchy problem (2.8) has a unique solution, which is represented by the following formula:
(3.12)uk=Rku0+∑s=1k‍Rk-s+1f(ts-1)(wts-wts-1),1≤k≤N.
Then from this formula and the multipoint nonlocal-boundary condition
(3.13)u0=∑j=1J‍αju[λj/τ]+φ(wλ1,…,wλJ),
we get
(3.14)u0=∑j=1J‍αjR[λj/τ]u0+∑j=1J‍αj∑s=1[λj/τ]‍R[λj/τ]-s+1f(ts-1)(wts-wts-1)+φ(wλ1,…,wλJ).
By Lemma 3.2 the operator I-∑j=1JαjR[λj/τ] has a bounded inverse Υτ=(I-∑j=1JαjR[λj/τ])-1. Then
(3.15)u0=Υτ{∑j=1J‍αj∑s=1[λj/τ]‍R[λj/τ]-s+1f(ts-1)(wts-wts-1)+φ(wλ1,…,wλJ)}.
Therefore, we have formulas (3.12) and (3.15) for the solution of problem (3.10). Now, we will study the convergence of difference scheme (3.10).Theorem 3.3.
Assume that(3.16)E∥A1/2φ(wλ1,…,wλJ)∥H2≤C.
Then the estimate of convergence
(3.17)max0≤k≤N(E∥v(tk)-uk∥H2)1/2≤C1(δ,λ1)τ1/2
holds. Here C and C1(δ,λ1) do not depend on τ.Proof.
Using formulas (2.12) and (3.15), we can write
(3.18)v(0)-u0=(Υ-Υτ)φ(wλ1,…,wλJ)+Υ∑j=1J‍αj∫0λje-A(λj-s)f(s)dws-Υτ∑j=1J‍αj∑s=1[λj/τ]‍R[λj/τ]-s+1f(ts-1)(wts-wts-1)=P1,J+P2,J+P3,J+P4,J+P5,J+P6,J+P7,J,
where
(3.19)P1,J=(Υ-Υτ)φ(wλ1,…,wλJ),(3.20)P2,J=(Υ-Υτ)∑j=1J‍αj∫0λje-A(λj-s)f(s)dws,(3.21)P3,J=Υτ∑j=1J‍αj∫[λj/τ]τλje-A(λj-s)f(s)dws,(3.22)P4,J=Υτ∑j=1J‍αj∑p=1[λj/τ]‍∫tp-1tp(e-(λj-s)A-e-([λj/τ]τ-s)A)f(s)dws,(3.23)P5,J=Υτ∑j=1J‍αj∑p=1[λj/τ]‍(e-([λj/τ]τ-pτ)A-R[λj/τ]-p)∫tp-1tpe-(tp-s)Af(s)dws,(3.24)P6,J=Υτ∑j=1J‍αj∑p=1[λj/τ]‍R[λj/τ]-p(∫tp-1tp∑p=1[λj/τ]‍e-(tp-s)Af(s)dws-∫tp-1tpe-τAf(tp-1)dws),(3.25)P7,J=Υτ∑j=1J‍αj∑p=1[λj/τ]‍R[λj/τ]-p(e-τA-R)f(tp-1)Δwtp.
Let us estimate Pk,J for all k=1,…,7, separately. We start with P1,J. Using formulas (2.4) and (3.4), we obtain
(3.26)Υ-Υτ=ΥΥτ(∑j=1J‍αj(e-Aλj-R[λj/τ])),
and also the expression in the above sum can be written in the following formula:
(3.27)e-Aλj-R[λj/τ]=-∫01d(R[λj/τ](x)e-(1-x)Aλj)=-∫01∑p=1[λj/τ]‍{[λjτ]R[λj/τ]-1(x)-τA(1+xτA)2+AλjR[λj/τ](x)}e-(1-x)Aλjdx=-∫01∑p=1[λj/τ]‍R[λj/τ]+1(x)e-(1-x)Aλj{-[λjτ]τA+(1+xτA)Aλj}dx=-∫01∑p=1[λj/τ]‍R[λj/τ]+1(x)e-(1-x)Aλj{(λj-[λjτ]τ)A+xτA2λj}dx.
Here R(x)=(I+τxA)-1. Using formulas (3.26), (3.27), and (3.19), we can write
(3.28)P1,J=ΥΥτ∑j=1J‍αj×(-∫01[λjτ]R[λj/τ]+1(x)e-(1-x)Aλj{(λj-[λjτ]τ)A+xτA2λj}dx)φ(wλ1,…,wλJ).
Let us estimate expected value of  P1,J. Since
(3.29)(λj-[λjτ]τ)≤τ,
we have that
(3.30)(E∥P1,J∥H2)1/2≤∥Υ∥H→H∥Υτ∥H→H×(E∥∑j=1J‍αjτ(∫01(wλ1,…,wλJ)A1/2R[λj/τ]+1(x)e-(1-x)AλjA1/2φ(wλ1,…,wλJ)dx)∥H2+λjτE∥∫01(wλ1,…,wλJ)A3/2R[λj/τ]+1(x)e-(1-x)AλjA1/2φ(wλ1,…,wλJ)dx∥H2)1/2.
In the same manner by using the triangle inequality and estimates (3.2) and (3.1), we get
(3.31)(E∥P1,J∥H2)1/2≤C1(δ,λ1)(∑j=1J‍∑j=1J‍|αj|∫01dx[λj/τ+1]τxτdx[λj/τ+1]τx×(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2+∑j=1J‍|αj|λj∫01/2τdx(1-x)3/2λj3/2×(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2+∑j=1J‍|αj|λj∫1/21dx[λj/τ+1]τxτdx([λj/τ]τx)3/2×(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2∑j=1J‍)≤C1(δ,λ1)(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2(∑j=1J‍C2τ1λj+∑j=1J‍C3τ1λj)≤C4(δ,λ1)∑j=1J‍C4τ1/21λj1/2(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2≤C5(δ,λ1)τ1/2(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2.
Now, let us estimate P2,J.  Using formula (3.20), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.32)(E∥P2,J∥H2)1/2≤C5(δ,λ1)τ1/2(∑j=1J‍E∥∫0λj∫0λj‍e-A(λj-s)A1/2f(s)dws∥H2)1/2≤C5(δ,λ1)τ1/2(∑j=1J‍∫0λj∫0λj‍∥A1/2f(s)∥H2ds)1/2≤C6(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Let us estimate P3,J. Using formula (3.21), the triangle inequality, and estimates (3.5), (3.2), and  (3.1), we get
(3.33)(E∥P3,J∥H2)1/2≤∥Υτ∥H→H∑j=1J‍|αj|(∫[λj/τ]τλj∥A1/2f(s)∥H2∥A1/2f(s)∥H2ds∥e-A(λj-s)∥H→H2∥A-1/2∥H→H2×∫[λj/τ]τλj∥A1/2f(s)∥H2∥A1/2f(s)∥H2ds)1/2,≤C6(δ,λ1)∑j=1J‍|αj|(∫[λj/τ]τλj∫0λj‍∥A1/2f(s)∥H2ds)1/2,≤C6(δ,λ1)∑j=1J‍|αj|(λj-[λjτ]τ)1/2max0≤s≤T(∥A1/2f(s)∥H2)1/2≤C7(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Next, let us estimate P4,J. Using formula (3.22), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.34)(E∥P4,J∥H2)1/2≤∥Υτ∥H→H∑j=1J‍|αj|×(∑p=1[λj/τ]‍∫tp-1tp∫0λj‍∥A-1/2(e-(λj-s)A-e-([λj/τ]τ-s)A)∥H→H2∥A1/2f(s)∥H2ds)1/2≤C1(δ,λ1)(∑p=1[λj/τ]‍τ∫tp-1tp∫0λj‍∥A1/2f(s)∥H2ds)1/2≤C1(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Next, let us estimate P5,J. Using formula (3.23), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.35)(E∥P5,J∥H2)1/2≤∥Υτ∥H→H(∑p=1[λj/τ]‍∑j=1J‍|αj|∑p=1[λj/τ]‍∥A-1/2(e-([λj/τ]τ-pτ)A-R[λj/τ]-p)∥H→H2×E∫tp-1tp∥e-(tp-s)A∥H→H2∥A1/2f(s)∥H2ds∑p=1[λj/τ]‍)1/2≤C2(δ,λ1)τ1/2(∫0T∥A1/2f(s)∥H2ds)1/2≤C2(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Next, let us estimate P6,J. Using formula (3.24), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.36)(E∥P6,J∥H2)1/2≤∥Υτ∥H→H∑j=1J‍|αj|×(∑p=1[λj/τ]‍∫tp-1tp∥R[λj/τ]-p∥H→H2∥R[λj/τ]-p∥H→H2E∥e-(tp-s)Af(s)-e-τAf(tp-1)∥H2ds)1/2≤C1(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍∫tp-1tp∫tp-1tp‍E∥e-(tp-s)Af(s)-e-(tp-tp-1)Af(tp-1)∥H2ds)1/2=C1(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍∑p=1[λj/τ]∫tp-1tpE∥(e-(tp-s)A-e-(tp-tp-1)A)f(s)‍+e-(tp-tp-1)A(f(s)-f(tp-1))∥H2ds∑p=1[λj/τ]‍)1/2≤C2(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍∫tp-1tp∫tp-1tp‍∥A-1/2(e-(tp-s)A-e-(tp-tp-1)A)A1/2f(s)∥H2+∥e-(tp-tp-1)A(f(s)-f(tp-1))∥H2ds∑p=1[λj/τ]‍)1/2≤C3(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍∫tp-1tp(τ∥A1/2f(s)∥H2+∥f(s)-f(tp-1)∥H2)ds)1/2≤C4(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍E∫tp-1tp∫0T‍(τ∥A1/2f(s)∥H2+∥f′(s)τ∥H2)ds)1/2≤C4(δ,λ1)τ1/2(∫0T∫0T‍∥A1/2f(s)∥H2ds+∫0T∫0T‍∥f′(s)∥H2ds)1/2≤C5(δ,λ1)τ1/2(max0≤s≤T∥A1/2f(s)∥H+max0≤s≤T∥f′(s)∥H).
Finally, let us estimate P7,J. Using formula (3.25), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.37)(E∥P5,J∥H2)1/2≤∥Υτ∥H→H∑j=1J‍|αj|×(∑p=1[λj/τ]‍∑p=1[λj/τ]‍∥R[λj/τ]-p∥H→H2∥A-1/2(e-τA-R)∥H→H2∥A1/2f(tp-1)∥H2×E∥Δwtp∥H2∑p=1[λj/τ]‍)1/2≤C(δ,λ1)(∑p=1[λj/τ]‍τ∥A1/2f(tp-1)∥H2E∥Δwtp∥H2)1/2.
Since Δwtp is a Wiener process and
(3.38)E∥Δwtp∥H2≤Δtp=τ,
we have that
(3.39)(E∥P5,J∥H2)1/2≤C(δ,λ1)τ1/2(∑p=1[λj/τ]‍∥A1/2f(tp-1)∥H2τ)1/2≤C1(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Applying estimates for Pk,J, k=1,…,7,  we get the estimate:
(3.40)(E∥v(t0)-u0∥H2)1/2≤C4(δ,λ1)τ1/2.
To prove the Theorem 3.3 it suffices to establish the following estimate:
(3.41)max1≤k≤N(E∥v(tk)-uk∥H2)1/2≤C2(δ,λ1)τ1/2.
Using formulas (2.9) and (3.12), we can write
(3.42)v(tk)-uk=e-kτAv(0)+∑s=1k‍e-(k-s)τA∫ts-1tse-A(ts-p)f(p)dwp-Rku0-∑s=1k‍Rk-s+1f(ts)(wts-wts-1)=P1,k+P2,k+P3,k+P4,k+P5,k,
where
(3.43)P1,k=(e-kτA-Rk)Υ{∑j=1J‍αj∫0λj∫ts-1ts‍e-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)‍},P2,k=Rk(v(0)-u0),P3,k=∑s=1k-1‍[e-(k-s)τA-Rk-s]∫ts-1ts∫‍e-A(ts-p)f(p)dwp,P4,k=∑s=1k‍Rk-s∫ts-1tse-A(ts-p)f(p)dwp-e-τAf(ts-1)(wts-wts-1),P5,k=∑s=1k‍Rk-s[e-τA-R]f(ts-1)(wts-wts-1).
Let us estimate Pm,k for all m=1,…,5, separately. We start with P1,k. Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get
(3.44)(E∥P1,k∥H2)1/2≤({∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)}∥(e-kτA-Rk)A-1/2∥H→H2×E∥A1/2Υ{∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)}∥H2{∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)})1/2≤C1(δ,λ1)τ1/2(E∥Υ∑j=1J‍αj∫0λje-A(λj-s)A1/2f(s)dws+A1/2φ(wλ1,…,wλJ)∑j=1J‍∥H2∑j=1J‍)1/2≤C2(δ,λ1)τ1/2∥Υ∥H→H((∑j=1J‍|αj|)(∑j=1J‍|αj|)2∫0λj∥e-A(λj-s)∥H→H2∥A1/2f(s)∥H2ds+∥A1/2φ(wλ1,…,wλJ)∥H2(∑j=1J‍|αj|))1/2≤C3(δ,λ1)τ1/2(∫0T∥A1/2f(s)∥H2ds+E∥A1/2φ∥H2)1/2≤C4(δ,λ1)τ1/2(max0≤s≤T∥A1/2f(s)∥H+(E∥A1/2φ∥H2)1/2).
Now, we estimate P2,k. Using estimate (3.1), we get
(3.45)(E∥P2,k∥H2)1/2≤(∥Rk∥H→H2E∥v(0)-u0∥H2)1/2≤(E∥v(0)-u0∥H2)1/2.
Applying the estimate (3.40), we obtain
(3.46)(E∥P2,k∥H2)1/2≤C(δ,λ1)τ1/2.
Now, we estimate P3,k. Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get
(3.47)(E∥P3,k∥H2)1/2≤C(δ,λ1)(∑s=1k-1‍∥A-1/2[e-(k-s)τA-Rk-s]∥H→H2×∥e-A(ts-p)∥H→H2∫ts-1ts∥A1/2f(p)∥H2dp)1/2≤C(δ,λ1)(∑s=1k-1‍τ∫ts-1ts∥A1/2f(p)∥H2dp)1/2≤C(δ,λ1)τ1/2(∫0T∥A1/2f(p)∥H2dp)1/2≤C(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Now, we estimate P4,k. We denote that
(3.48)bj=∫tj-1tj∫tj-1s(e-τAA-1/2f′(z)+A1/2e-(tj-z)Af(s))dzdws,bj*={bj,1≤j≤k-1,0,otherwise.
Then
(3.49)P4,k=∑s=1k‍Rk-s∫ts-1ts∫tj-1tj‍(e-A(ts-p)f(p)-e-τAf(ts-1))dwp=∑s=1k‍Rk-s∫ts-1ts((e-A(ts-p)-e-τA)f(p)+e-τA(f(p)-f(ts-1)))dwp=∑s=1k‍A1/2Rk-s∫tj-1tj∫tj-1s(e-τAA-1/2f′(z)+A1/2e-(tj-z)Af(s))dzdws=∑i=1N‍RiA1/2bk-i*,(E∥P4,k∥H2)1/2=(E∥∑i=1N‍RiA1/2bk-i*∥H2)1/2.
Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get
(3.50)(E∥P4,k∥2)1/2≤(∑i=1N‍E∥RiA1/2bk-i*∥H2)1/2≤(∑i=1N‍∥A1/2Ri∥H→HE∥bk-i*∥H2)1/2.
Since
(3.51)(E∥bj*∥H2)1/2=(E∥∫tj-1tj∫tj-1s∫tj-1tj‍(e-τAA-1/2f′(z)+A1/2e-(tj-z)Af(s))dzdws∥H2)1/2≤∫tj-1tj(∫tj-1s∥(e-τAA-1/2f′(z)+A1/2e-(tj-zA)f(s))∥H2dz)1/2ds≤∫tj-1tj∫tj-1s∫tj-1tj‍∥(e-τAA-1/2f′(z)+A1/2e-(tj-z)Af(s))∥H2dzds≤C(δ,λ1)τ3/2(max0≤s≤T∥A1/2f(s)∥H+max0≤s≤T∥A-1/2f′(s)∥H),
we have that
(3.52)(E∥P4,k∥H2)1/2≤∑i=1N‍C1iτ(E∥bk-i*∥H2)1/2≤∑i=1N‍C1iτC(δ,λ1)τ3/2(max0≤s≤T∥A1/2f(s)∥H+max0≤s≤T∥A-1/2f′(s)∥H)≤C1(δ,λ1)τ1/2(max0≤s≤T∥A1/2f(s)∥H+max0≤s≤T∥A-1/2f′(s)∥H).
Finally, we estimate P5,k. We denote that
(3.53)qj=A1/2f(tj-1)Δwtj,aj*={qj,1≤j≤k-1,0,otherwise.
Therefore,
(3.54)P5,k=∑i=1N‍A1/2RiA-1(e-τA-R)qk-i*.
Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get
(3.55)(E∥P5,k∥H2)1/2≤∑i=1N‍∥A1/2Ri∥H→H∥A-1(e-τA-R)∥H→H(E∥qk-i*∥H2)1/2≤∑i=1N‍2τiτ(E∥qk-i*∥H2)1/2≤Cmax1≤j≤N(E∥qj∥H2)1/2.
Since
(3.56)(E∥qj∥H2)1/2≤(E∥A1/2f(tj-1)Δwtj∥H2)1/2≤C1(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H,
we have that
(3.57)(E∥P5,k∥H2)1/2≤C2(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Combining estimates P1,k,P2,k,P3,k,P4,k, and P5,k, we obtain (3.41). Theorem 3.3 is proved.


### 3.2. Applications

Now, we consider applications of Theorem3.3. First, let us consider the nonlocal-boundary value problem for one-dimensional stochastic parabolic equation:
(3.58)du(t,x)-(a(x)ux)xdt+δu(t,x)dt=f(t,x)dwt,0<t<T,0<x<1,u(0,x)=∑j=1J‍αju(λj,x)+φ(wλ1,…,wλJ,x),0≤x≤1,∑j=1J‍|αj|≤1,0<λ1<⋯<λJ≤T,wt=tξ,ξ∈N(0,1),0≤t≤T,u(t,0)=u(t,1),ux(t,0)=ux(t,1),0≤t≤T,
where δ>0,a(x)≥a>0(x∈(0,1)),φ(wλ1,…,wλJ,x)(x∈[0,1]) and f(t,x)(t,x∈[0,1]) are smooth functions with respect to x.The discretization of problem (3.58) is carried out in two steps. In the first step, we define the grid space
(3.59)[0,1]h={x=xn:xn=nh,0≤n≤M,Mh=1}.Let us introduce the Hilbert spaceL2h=L2([0,1]h) of the grid functions φh(x)={φn}1M-1 defined on [0,1]h, equipped with the norm
(3.60)∥φh∥L2h=(∑x∈[0,1]h‍|φ(x)|2h)1/2.To the differential operatorA generated by problem (3.58), we assign the difference operator Ahx by the formula
(3.61)Ahxφh(x)={-(a(x)φx¯)x,n+δφn}1M-1
acting in the space of grid functions φh(x)={φn}0M satisfying the conditions φ0=φM, φ1-φ0=φM-φM-1. It is well known that Ahx is a self-adjoint positive definite operator in L2h. With the help of Ahx, we arrive at the nonlocal-boundary value problem:
(3.62)duh(t,x)+Ahxuh(t,x)dt=fh(t,x)dwt,0<t<T,x∈[0,1]h,uh(0,x)=∑j=1J‍αjuh(λj,x)+φ(wλ1,…,wλJ,x),x∈[0,1]h.In the second step, we replace (3.62) with the difference scheme (3.10):
(3.63)ukh(x)-uk-1h(x)+τAhxukh(x)=fk-1h(x)(wtk-wtk-1),1≤k≤N,fk-1h(x)=fh(tk-1,x),tk=kτ,1≤k≤N,x∈[0,1]h,u0h(x)=∑j=1J‍αju[λj/τ]h(x)+φ(wλ1,…,wλJ,x),x∈[0,1]h.Theorem 3.4.
Letτ and h be sufficiently small positive numbers. Then, the solutions of difference scheme (3.63) satisfy the following convergence estimate:
(3.64)max0≤k≤N(E∥u(tk)-ukh∥L2h2)1/2≤C(δ,λ1)(τ1/2+h),
where C(δ,λ1) do not depend on τ and h. Here, one puts u(tk)={u(tk,xn)}0M as the grid function of exact solution of problem (3.58) at the grid points t=tk,0≤k≤N and x=xn,0≤n≤M.Proof.
Let us introduce the Banach spaceC([0,1],H) of abstract mesh functions uk=ukh(x) defined on [0,1]τ with values in H=L2h. Then, difference scheme (3.63) can be reduced to the abstract difference scheme:
(3.65)(uk-uk-1)+τAuk=fk,fk=f(tk-1),tk=kτ,1≤k≤N,u0=∑j=1J‍αju[λj/τ]+φ(wλ1,…,wλJ),
in a Hilbert space L2h with the operator A=Ahx by formula (3.62). It is clear that A=A* and A≥δI in H=L2h. Hence, Ahx is a self-adjoint positive definite operator in L2h. Therefore, Theorem 3.3 applies to this case, and Theorem 3.4 is proved.Second, letΩ be the unit open cube in the n-dimensional Euclidean space ℝn={x=(x1,…,xn):0<xi<1,i=1,…,n} with boundary S,Ω¯=Ω∪S. In [0,T]×Ω, the nonlocal boundary value problem for the multidimensional parabolic equation
(3.66)du(t,x)-∑r=1n‍(ar(x)uxr)xrdt=f(t,x)dwt,0<t<T,x=(x1,…,xn)∈Ω,u(0,x)=∑j=1J‍αju(λj,x)+φ(wλ1,…,wλJ,x),x∈Ω¯,∑j=1J‍|αj|≤1,0<λ1<⋯<λJ≤T,wt=tξ,ξ∈N(0,1),0≤t≤T,u(t,x)=0,x∈S,0≤t≤T
with the Dirichlet condition is considered. Here ar(x),(x∈Ω),φ(x)(x∈Ω¯), and f(t,x)(t∈(0,1), x∈Ω) are given smooth functions with respect to x and ar(x)≥a>0.The discretization of problem (3.66) is carried out in two steps. In the first step, define the grid space Ω~h={x=xm=(h1m1,…,hnmn);m=(m1,…,mn),0≤mr≤Nr,hrNr=1,r=1,…,n},Ωh=Ω~h∩Ω,Sh=Ω~h∩S.LetL2h denote the Hilbert space
(3.67)L2h=L2(Ω~h)={φh(x):(∑x∈Ω~h‍|φh(x)|2h1⋯hn)1/2<∞}.
The differential operator A in (3.66) is replaced with
(3.68)Ahxuh(x)=-∑r=1n‍(ar(x)ux¯rh)xr,jr,
where the difference operator Ahx is defined on those grid functions uh(x)=0, for all x∈Sh. It is well known that Ahx is a self-adjoint positive definite operator in L2h.Using (3.66) and (3.68), we get
(3.69)duh(t,x)+Ahxuh(t,x)dt=fh(t,x)dwt,0<t<T,x∈Ωh,uh(0,x)=∑j=1J‍αjuh(λj,x)+φ(wλ1,…,wλJ,x),x∈Ω~h.In the second step, we replace (3.69) with the difference scheme (3.10):
(3.70)ukh(x)-uk-1h(x)+τAhxukh(x)=fk-1h(x)(wtk-wtk-1),1≤k≤N,fk-1h(x)=fh(tk-1,x),tk=kτ,1≤k≤N,x∈Ωh,u0h(x)=∑j=1J‍αju[λj/τ]h(x)+φ(wλ1,…,wλJ,x),x∈Ω~h.Theorem 3.5.
Letτ and |h|=h12+⋯+hn2 be sufficiently small positive numbers. Then, the solution of difference scheme (3.70) satisfies the following convergence estimate:
(3.71)max0≤k≤N(E∥u(tk)-ukh∥L2h2)1/2≤C(δ,λ1)(τ1/2+|h|2),
where C(δ,λ1) do not depend on τ and |h|. Here, one puts u(tk)=u(tk,x)|x∈Ω~h as the grid function of exact solution of problem (3.66) at the grid points t=tk,0≤k≤N and x∈Ω~h.The proof of Theorem3.5 is based on the abstract Theorem 3.3 and the symmetry properties of the difference operator Ahx defined by formula (3.68).


## 3.1. 1/2-th Order-of-Accuracy Rothe Difference Scheme

Let us give some lemmas we need in the sequel.Lemma 3.1.
The following estimates hold:(3.1)∥AαRk∥H→H≤1(kτ)α,1≤k≤N,0≤α≤1,(3.2)∥A-α(Rk-e-kτA)∥H→H≤2ταk1-α,1≤k≤N,0≤α≤2,
where R=(I+τA)-1.Lemma 3.2.
Suppose that assumption (2.2) holds. Then, the operator
(3.3)I-∑j=1J‍αjR[λj/τ]
has a bounded inverse
(3.4)Υτ=(I-∑j=1J‍αjR[λj/τ])-1,
and the following estimate is satisfied:
(3.5)∥Υτ∥H→H≤C(δ,λ1).Proof.
The proof follows from the triangle inequality, assumption (2.2), and estimate
(3.6)∥(I-∑j=1J‍αjR[λj/τ])-1∥H→H≤supδ≤μ<∞1|1-∑j=1Jαk(1/(1+μτ)[λk/τ])|.
From (2.14) it is clear that for the approximate solution of the multipoint nonlocal-boundary value problem (1.1) it is necessary to approximate the expressions
(3.7)e-τA,1τ∫tk-1tke-(tk-s)Af(s)dw,
and multipoint nonlocal-boundary condition
(3.8)v(0)=∑j=1J‍αjv(λj)+φ(wλ1,…,wλJ).
It is possible under stronger assumption than (ii) forf(t). Assume that
(3.9)maxt∈[0,T]∥A-1/2f′(t)∥H+maxt∈[0,T]∥A1/2f(t)∥H≤C.
Replacing the expressions e-τA,e-(tk-s)A with R=(I+τA)-1, the expression v(λj) with v([λj/τ]τ), and the function f(s) with f(tk-1), we get the implicit Rothe difference scheme:
(3.10)uk-uk-1+τAuk=f(tk-1)(wtk-wtk-1),1≤k≤N,u0=∑j=1J‍αju[λj/τ]+φ(wλ1,…,wλJ).
Let us now obtain the formula for the solution of problem (3.10). It is clear that the Rothe difference scheme
(3.11)uk-uk-1+τAuk=f(tk-1)(wtk-wtk-1),1≤k≤N,u0isgiven,
for the solution of the Cauchy problem (2.8) has a unique solution, which is represented by the following formula:
(3.12)uk=Rku0+∑s=1k‍Rk-s+1f(ts-1)(wts-wts-1),1≤k≤N.
Then from this formula and the multipoint nonlocal-boundary condition
(3.13)u0=∑j=1J‍αju[λj/τ]+φ(wλ1,…,wλJ),
we get
(3.14)u0=∑j=1J‍αjR[λj/τ]u0+∑j=1J‍αj∑s=1[λj/τ]‍R[λj/τ]-s+1f(ts-1)(wts-wts-1)+φ(wλ1,…,wλJ).
By Lemma 3.2 the operator I-∑j=1JαjR[λj/τ] has a bounded inverse Υτ=(I-∑j=1JαjR[λj/τ])-1. Then
(3.15)u0=Υτ{∑j=1J‍αj∑s=1[λj/τ]‍R[λj/τ]-s+1f(ts-1)(wts-wts-1)+φ(wλ1,…,wλJ)}.
Therefore, we have formulas (3.12) and (3.15) for the solution of problem (3.10). Now, we will study the convergence of difference scheme (3.10).Theorem 3.3.
Assume that(3.16)E∥A1/2φ(wλ1,…,wλJ)∥H2≤C.
Then the estimate of convergence
(3.17)max0≤k≤N(E∥v(tk)-uk∥H2)1/2≤C1(δ,λ1)τ1/2
holds. Here C and C1(δ,λ1) do not depend on τ.Proof.
Using formulas (2.12) and (3.15), we can write
(3.18)v(0)-u0=(Υ-Υτ)φ(wλ1,…,wλJ)+Υ∑j=1J‍αj∫0λje-A(λj-s)f(s)dws-Υτ∑j=1J‍αj∑s=1[λj/τ]‍R[λj/τ]-s+1f(ts-1)(wts-wts-1)=P1,J+P2,J+P3,J+P4,J+P5,J+P6,J+P7,J,
where
(3.19)P1,J=(Υ-Υτ)φ(wλ1,…,wλJ),(3.20)P2,J=(Υ-Υτ)∑j=1J‍αj∫0λje-A(λj-s)f(s)dws,(3.21)P3,J=Υτ∑j=1J‍αj∫[λj/τ]τλje-A(λj-s)f(s)dws,(3.22)P4,J=Υτ∑j=1J‍αj∑p=1[λj/τ]‍∫tp-1tp(e-(λj-s)A-e-([λj/τ]τ-s)A)f(s)dws,(3.23)P5,J=Υτ∑j=1J‍αj∑p=1[λj/τ]‍(e-([λj/τ]τ-pτ)A-R[λj/τ]-p)∫tp-1tpe-(tp-s)Af(s)dws,(3.24)P6,J=Υτ∑j=1J‍αj∑p=1[λj/τ]‍R[λj/τ]-p(∫tp-1tp∑p=1[λj/τ]‍e-(tp-s)Af(s)dws-∫tp-1tpe-τAf(tp-1)dws),(3.25)P7,J=Υτ∑j=1J‍αj∑p=1[λj/τ]‍R[λj/τ]-p(e-τA-R)f(tp-1)Δwtp.
Let us estimate Pk,J for all k=1,…,7, separately. We start with P1,J. Using formulas (2.4) and (3.4), we obtain
(3.26)Υ-Υτ=ΥΥτ(∑j=1J‍αj(e-Aλj-R[λj/τ])),
and also the expression in the above sum can be written in the following formula:
(3.27)e-Aλj-R[λj/τ]=-∫01d(R[λj/τ](x)e-(1-x)Aλj)=-∫01∑p=1[λj/τ]‍{[λjτ]R[λj/τ]-1(x)-τA(1+xτA)2+AλjR[λj/τ](x)}e-(1-x)Aλjdx=-∫01∑p=1[λj/τ]‍R[λj/τ]+1(x)e-(1-x)Aλj{-[λjτ]τA+(1+xτA)Aλj}dx=-∫01∑p=1[λj/τ]‍R[λj/τ]+1(x)e-(1-x)Aλj{(λj-[λjτ]τ)A+xτA2λj}dx.
Here R(x)=(I+τxA)-1. Using formulas (3.26), (3.27), and (3.19), we can write
(3.28)P1,J=ΥΥτ∑j=1J‍αj×(-∫01[λjτ]R[λj/τ]+1(x)e-(1-x)Aλj{(λj-[λjτ]τ)A+xτA2λj}dx)φ(wλ1,…,wλJ).
Let us estimate expected value of  P1,J. Since
(3.29)(λj-[λjτ]τ)≤τ,
we have that
(3.30)(E∥P1,J∥H2)1/2≤∥Υ∥H→H∥Υτ∥H→H×(E∥∑j=1J‍αjτ(∫01(wλ1,…,wλJ)A1/2R[λj/τ]+1(x)e-(1-x)AλjA1/2φ(wλ1,…,wλJ)dx)∥H2+λjτE∥∫01(wλ1,…,wλJ)A3/2R[λj/τ]+1(x)e-(1-x)AλjA1/2φ(wλ1,…,wλJ)dx∥H2)1/2.
In the same manner by using the triangle inequality and estimates (3.2) and (3.1), we get
(3.31)(E∥P1,J∥H2)1/2≤C1(δ,λ1)(∑j=1J‍∑j=1J‍|αj|∫01dx[λj/τ+1]τxτdx[λj/τ+1]τx×(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2+∑j=1J‍|αj|λj∫01/2τdx(1-x)3/2λj3/2×(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2+∑j=1J‍|αj|λj∫1/21dx[λj/τ+1]τxτdx([λj/τ]τx)3/2×(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2∑j=1J‍)≤C1(δ,λ1)(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2(∑j=1J‍C2τ1λj+∑j=1J‍C3τ1λj)≤C4(δ,λ1)∑j=1J‍C4τ1/21λj1/2(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2≤C5(δ,λ1)τ1/2(E∥A1/2φ(wλ1,…,wλJ)∥H2)1/2.
Now, let us estimate P2,J.  Using formula (3.20), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.32)(E∥P2,J∥H2)1/2≤C5(δ,λ1)τ1/2(∑j=1J‍E∥∫0λj∫0λj‍e-A(λj-s)A1/2f(s)dws∥H2)1/2≤C5(δ,λ1)τ1/2(∑j=1J‍∫0λj∫0λj‍∥A1/2f(s)∥H2ds)1/2≤C6(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Let us estimate P3,J. Using formula (3.21), the triangle inequality, and estimates (3.5), (3.2), and  (3.1), we get
(3.33)(E∥P3,J∥H2)1/2≤∥Υτ∥H→H∑j=1J‍|αj|(∫[λj/τ]τλj∥A1/2f(s)∥H2∥A1/2f(s)∥H2ds∥e-A(λj-s)∥H→H2∥A-1/2∥H→H2×∫[λj/τ]τλj∥A1/2f(s)∥H2∥A1/2f(s)∥H2ds)1/2,≤C6(δ,λ1)∑j=1J‍|αj|(∫[λj/τ]τλj∫0λj‍∥A1/2f(s)∥H2ds)1/2,≤C6(δ,λ1)∑j=1J‍|αj|(λj-[λjτ]τ)1/2max0≤s≤T(∥A1/2f(s)∥H2)1/2≤C7(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Next, let us estimate P4,J. Using formula (3.22), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.34)(E∥P4,J∥H2)1/2≤∥Υτ∥H→H∑j=1J‍|αj|×(∑p=1[λj/τ]‍∫tp-1tp∫0λj‍∥A-1/2(e-(λj-s)A-e-([λj/τ]τ-s)A)∥H→H2∥A1/2f(s)∥H2ds)1/2≤C1(δ,λ1)(∑p=1[λj/τ]‍τ∫tp-1tp∫0λj‍∥A1/2f(s)∥H2ds)1/2≤C1(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Next, let us estimate P5,J. Using formula (3.23), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.35)(E∥P5,J∥H2)1/2≤∥Υτ∥H→H(∑p=1[λj/τ]‍∑j=1J‍|αj|∑p=1[λj/τ]‍∥A-1/2(e-([λj/τ]τ-pτ)A-R[λj/τ]-p)∥H→H2×E∫tp-1tp∥e-(tp-s)A∥H→H2∥A1/2f(s)∥H2ds∑p=1[λj/τ]‍)1/2≤C2(δ,λ1)τ1/2(∫0T∥A1/2f(s)∥H2ds)1/2≤C2(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Next, let us estimate P6,J. Using formula (3.24), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.36)(E∥P6,J∥H2)1/2≤∥Υτ∥H→H∑j=1J‍|αj|×(∑p=1[λj/τ]‍∫tp-1tp∥R[λj/τ]-p∥H→H2∥R[λj/τ]-p∥H→H2E∥e-(tp-s)Af(s)-e-τAf(tp-1)∥H2ds)1/2≤C1(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍∫tp-1tp∫tp-1tp‍E∥e-(tp-s)Af(s)-e-(tp-tp-1)Af(tp-1)∥H2ds)1/2=C1(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍∑p=1[λj/τ]∫tp-1tpE∥(e-(tp-s)A-e-(tp-tp-1)A)f(s)‍+e-(tp-tp-1)A(f(s)-f(tp-1))∥H2ds∑p=1[λj/τ]‍)1/2≤C2(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍∫tp-1tp∫tp-1tp‍∥A-1/2(e-(tp-s)A-e-(tp-tp-1)A)A1/2f(s)∥H2+∥e-(tp-tp-1)A(f(s)-f(tp-1))∥H2ds∑p=1[λj/τ]‍)1/2≤C3(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍∫tp-1tp(τ∥A1/2f(s)∥H2+∥f(s)-f(tp-1)∥H2)ds)1/2≤C4(δ,λ1)∑j=1J‍(∑p=1[λj/τ]‍E∫tp-1tp∫0T‍(τ∥A1/2f(s)∥H2+∥f′(s)τ∥H2)ds)1/2≤C4(δ,λ1)τ1/2(∫0T∫0T‍∥A1/2f(s)∥H2ds+∫0T∫0T‍∥f′(s)∥H2ds)1/2≤C5(δ,λ1)τ1/2(max0≤s≤T∥A1/2f(s)∥H+max0≤s≤T∥f′(s)∥H).
Finally, let us estimate P7,J. Using formula (3.25), the triangle inequality, and estimates (3.5), (3.2), and (3.1), we get
(3.37)(E∥P5,J∥H2)1/2≤∥Υτ∥H→H∑j=1J‍|αj|×(∑p=1[λj/τ]‍∑p=1[λj/τ]‍∥R[λj/τ]-p∥H→H2∥A-1/2(e-τA-R)∥H→H2∥A1/2f(tp-1)∥H2×E∥Δwtp∥H2∑p=1[λj/τ]‍)1/2≤C(δ,λ1)(∑p=1[λj/τ]‍τ∥A1/2f(tp-1)∥H2E∥Δwtp∥H2)1/2.
Since Δwtp is a Wiener process and
(3.38)E∥Δwtp∥H2≤Δtp=τ,
we have that
(3.39)(E∥P5,J∥H2)1/2≤C(δ,λ1)τ1/2(∑p=1[λj/τ]‍∥A1/2f(tp-1)∥H2τ)1/2≤C1(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Applying estimates for Pk,J, k=1,…,7,  we get the estimate:
(3.40)(E∥v(t0)-u0∥H2)1/2≤C4(δ,λ1)τ1/2.
To prove the Theorem 3.3 it suffices to establish the following estimate:
(3.41)max1≤k≤N(E∥v(tk)-uk∥H2)1/2≤C2(δ,λ1)τ1/2.
Using formulas (2.9) and (3.12), we can write
(3.42)v(tk)-uk=e-kτAv(0)+∑s=1k‍e-(k-s)τA∫ts-1tse-A(ts-p)f(p)dwp-Rku0-∑s=1k‍Rk-s+1f(ts)(wts-wts-1)=P1,k+P2,k+P3,k+P4,k+P5,k,
where
(3.43)P1,k=(e-kτA-Rk)Υ{∑j=1J‍αj∫0λj∫ts-1ts‍e-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)‍},P2,k=Rk(v(0)-u0),P3,k=∑s=1k-1‍[e-(k-s)τA-Rk-s]∫ts-1ts∫‍e-A(ts-p)f(p)dwp,P4,k=∑s=1k‍Rk-s∫ts-1tse-A(ts-p)f(p)dwp-e-τAf(ts-1)(wts-wts-1),P5,k=∑s=1k‍Rk-s[e-τA-R]f(ts-1)(wts-wts-1).
Let us estimate Pm,k for all m=1,…,5, separately. We start with P1,k. Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get
(3.44)(E∥P1,k∥H2)1/2≤({∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)}∥(e-kτA-Rk)A-1/2∥H→H2×E∥A1/2Υ{∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)}∥H2{∑j=1J‍αj∫0λje-A(λj-s)f(s)dws+φ(wλ1,…,wλJ)})1/2≤C1(δ,λ1)τ1/2(E∥Υ∑j=1J‍αj∫0λje-A(λj-s)A1/2f(s)dws+A1/2φ(wλ1,…,wλJ)∑j=1J‍∥H2∑j=1J‍)1/2≤C2(δ,λ1)τ1/2∥Υ∥H→H((∑j=1J‍|αj|)(∑j=1J‍|αj|)2∫0λj∥e-A(λj-s)∥H→H2∥A1/2f(s)∥H2ds+∥A1/2φ(wλ1,…,wλJ)∥H2(∑j=1J‍|αj|))1/2≤C3(δ,λ1)τ1/2(∫0T∥A1/2f(s)∥H2ds+E∥A1/2φ∥H2)1/2≤C4(δ,λ1)τ1/2(max0≤s≤T∥A1/2f(s)∥H+(E∥A1/2φ∥H2)1/2).
Now, we estimate P2,k. Using estimate (3.1), we get
(3.45)(E∥P2,k∥H2)1/2≤(∥Rk∥H→H2E∥v(0)-u0∥H2)1/2≤(E∥v(0)-u0∥H2)1/2.
Applying the estimate (3.40), we obtain
(3.46)(E∥P2,k∥H2)1/2≤C(δ,λ1)τ1/2.
Now, we estimate P3,k. Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get
(3.47)(E∥P3,k∥H2)1/2≤C(δ,λ1)(∑s=1k-1‍∥A-1/2[e-(k-s)τA-Rk-s]∥H→H2×∥e-A(ts-p)∥H→H2∫ts-1ts∥A1/2f(p)∥H2dp)1/2≤C(δ,λ1)(∑s=1k-1‍τ∫ts-1ts∥A1/2f(p)∥H2dp)1/2≤C(δ,λ1)τ1/2(∫0T∥A1/2f(p)∥H2dp)1/2≤C(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Now, we estimate P4,k. We denote that
(3.48)bj=∫tj-1tj∫tj-1s(e-τAA-1/2f′(z)+A1/2e-(tj-z)Af(s))dzdws,bj*={bj,1≤j≤k-1,0,otherwise.
Then
(3.49)P4,k=∑s=1k‍Rk-s∫ts-1ts∫tj-1tj‍(e-A(ts-p)f(p)-e-τAf(ts-1))dwp=∑s=1k‍Rk-s∫ts-1ts((e-A(ts-p)-e-τA)f(p)+e-τA(f(p)-f(ts-1)))dwp=∑s=1k‍A1/2Rk-s∫tj-1tj∫tj-1s(e-τAA-1/2f′(z)+A1/2e-(tj-z)Af(s))dzdws=∑i=1N‍RiA1/2bk-i*,(E∥P4,k∥H2)1/2=(E∥∑i=1N‍RiA1/2bk-i*∥H2)1/2.
Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get
(3.50)(E∥P4,k∥2)1/2≤(∑i=1N‍E∥RiA1/2bk-i*∥H2)1/2≤(∑i=1N‍∥A1/2Ri∥H→HE∥bk-i*∥H2)1/2.
Since
(3.51)(E∥bj*∥H2)1/2=(E∥∫tj-1tj∫tj-1s∫tj-1tj‍(e-τAA-1/2f′(z)+A1/2e-(tj-z)Af(s))dzdws∥H2)1/2≤∫tj-1tj(∫tj-1s∥(e-τAA-1/2f′(z)+A1/2e-(tj-zA)f(s))∥H2dz)1/2ds≤∫tj-1tj∫tj-1s∫tj-1tj‍∥(e-τAA-1/2f′(z)+A1/2e-(tj-z)Af(s))∥H2dzds≤C(δ,λ1)τ3/2(max0≤s≤T∥A1/2f(s)∥H+max0≤s≤T∥A-1/2f′(s)∥H),
we have that
(3.52)(E∥P4,k∥H2)1/2≤∑i=1N‍C1iτ(E∥bk-i*∥H2)1/2≤∑i=1N‍C1iτC(δ,λ1)τ3/2(max0≤s≤T∥A1/2f(s)∥H+max0≤s≤T∥A-1/2f′(s)∥H)≤C1(δ,λ1)τ1/2(max0≤s≤T∥A1/2f(s)∥H+max0≤s≤T∥A-1/2f′(s)∥H).
Finally, we estimate P5,k. We denote that
(3.53)qj=A1/2f(tj-1)Δwtj,aj*={qj,1≤j≤k-1,0,otherwise.
Therefore,
(3.54)P5,k=∑i=1N‍A1/2RiA-1(e-τA-R)qk-i*.
Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get
(3.55)(E∥P5,k∥H2)1/2≤∑i=1N‍∥A1/2Ri∥H→H∥A-1(e-τA-R)∥H→H(E∥qk-i*∥H2)1/2≤∑i=1N‍2τiτ(E∥qk-i*∥H2)1/2≤Cmax1≤j≤N(E∥qj∥H2)1/2.
Since
(3.56)(E∥qj∥H2)1/2≤(E∥A1/2f(tj-1)Δwtj∥H2)1/2≤C1(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H,
we have that
(3.57)(E∥P5,k∥H2)1/2≤C2(δ,λ1)τ1/2max0≤s≤T∥A1/2f(s)∥H.
Combining estimates P1,k,P2,k,P3,k,P4,k, and P5,k, we obtain (3.41). Theorem 3.3 is proved.


## 3.2. Applications

Now, we consider applications of Theorem3.3. First, let us consider the nonlocal-boundary value problem for one-dimensional stochastic parabolic equation:
(3.58)du(t,x)-(a(x)ux)xdt+δu(t,x)dt=f(t,x)dwt,0<t<T,0<x<1,u(0,x)=∑j=1J‍αju(λj,x)+φ(wλ1,…,wλJ,x),0≤x≤1,∑j=1J‍|αj|≤1,0<λ1<⋯<λJ≤T,wt=tξ,ξ∈N(0,1),0≤t≤T,u(t,0)=u(t,1),ux(t,0)=ux(t,1),0≤t≤T,
where δ>0,a(x)≥a>0(x∈(0,1)),φ(wλ1,…,wλJ,x)(x∈[0,1]) and f(t,x)(t,x∈[0,1]) are smooth functions with respect to x.The discretization of problem (3.58) is carried out in two steps. In the first step, we define the grid space
(3.59)[0,1]h={x=xn:xn=nh,0≤n≤M,Mh=1}.Let us introduce the Hilbert spaceL2h=L2([0,1]h) of the grid functions φh(x)={φn}1M-1 defined on [0,1]h, equipped with the norm
(3.60)∥φh∥L2h=(∑x∈[0,1]h‍|φ(x)|2h)1/2.To the differential operatorA generated by problem (3.58), we assign the difference operator Ahx by the formula
(3.61)Ahxφh(x)={-(a(x)φx¯)x,n+δφn}1M-1
acting in the space of grid functions φh(x)={φn}0M satisfying the conditions φ0=φM, φ1-φ0=φM-φM-1. It is well known that Ahx is a self-adjoint positive definite operator in L2h. With the help of Ahx, we arrive at the nonlocal-boundary value problem:
(3.62)duh(t,x)+Ahxuh(t,x)dt=fh(t,x)dwt,0<t<T,x∈[0,1]h,uh(0,x)=∑j=1J‍αjuh(λj,x)+φ(wλ1,…,wλJ,x),x∈[0,1]h.In the second step, we replace (3.62) with the difference scheme (3.10):
(3.63)ukh(x)-uk-1h(x)+τAhxukh(x)=fk-1h(x)(wtk-wtk-1),1≤k≤N,fk-1h(x)=fh(tk-1,x),tk=kτ,1≤k≤N,x∈[0,1]h,u0h(x)=∑j=1J‍αju[λj/τ]h(x)+φ(wλ1,…,wλJ,x),x∈[0,1]h.Theorem 3.4.
Letτ and h be sufficiently small positive numbers. Then, the solutions of difference scheme (3.63) satisfy the following convergence estimate:
(3.64)max0≤k≤N(E∥u(tk)-ukh∥L2h2)1/2≤C(δ,λ1)(τ1/2+h),
where C(δ,λ1) do not depend on τ and h. Here, one puts u(tk)={u(tk,xn)}0M as the grid function of exact solution of problem (3.58) at the grid points t=tk,0≤k≤N and x=xn,0≤n≤M.Proof.
Let us introduce the Banach spaceC([0,1],H) of abstract mesh functions uk=ukh(x) defined on [0,1]τ with values in H=L2h. Then, difference scheme (3.63) can be reduced to the abstract difference scheme:
(3.65)(uk-uk-1)+τAuk=fk,fk=f(tk-1),tk=kτ,1≤k≤N,u0=∑j=1J‍αju[λj/τ]+φ(wλ1,…,wλJ),
in a Hilbert space L2h with the operator A=Ahx by formula (3.62). It is clear that A=A* and A≥δI in H=L2h. Hence, Ahx is a self-adjoint positive definite operator in L2h. Therefore, Theorem 3.3 applies to this case, and Theorem 3.4 is proved.Second, letΩ be the unit open cube in the n-dimensional Euclidean space ℝn={x=(x1,…,xn):0<xi<1,i=1,…,n} with boundary S,Ω¯=Ω∪S. In [0,T]×Ω, the nonlocal boundary value problem for the multidimensional parabolic equation
(3.66)du(t,x)-∑r=1n‍(ar(x)uxr)xrdt=f(t,x)dwt,0<t<T,x=(x1,…,xn)∈Ω,u(0,x)=∑j=1J‍αju(λj,x)+φ(wλ1,…,wλJ,x),x∈Ω¯,∑j=1J‍|αj|≤1,0<λ1<⋯<λJ≤T,wt=tξ,ξ∈N(0,1),0≤t≤T,u(t,x)=0,x∈S,0≤t≤T
with the Dirichlet condition is considered. Here ar(x),(x∈Ω),φ(x)(x∈Ω¯), and f(t,x)(t∈(0,1), x∈Ω) are given smooth functions with respect to x and ar(x)≥a>0.The discretization of problem (3.66) is carried out in two steps. In the first step, define the grid space Ω~h={x=xm=(h1m1,…,hnmn);m=(m1,…,mn),0≤mr≤Nr,hrNr=1,r=1,…,n},Ωh=Ω~h∩Ω,Sh=Ω~h∩S.LetL2h denote the Hilbert space
(3.67)L2h=L2(Ω~h)={φh(x):(∑x∈Ω~h‍|φh(x)|2h1⋯hn)1/2<∞}.
The differential operator A in (3.66) is replaced with
(3.68)Ahxuh(x)=-∑r=1n‍(ar(x)ux¯rh)xr,jr,
where the difference operator Ahx is defined on those grid functions uh(x)=0, for all x∈Sh. It is well known that Ahx is a self-adjoint positive definite operator in L2h.Using (3.66) and (3.68), we get
(3.69)duh(t,x)+Ahxuh(t,x)dt=fh(t,x)dwt,0<t<T,x∈Ωh,uh(0,x)=∑j=1J‍αjuh(λj,x)+φ(wλ1,…,wλJ,x),x∈Ω~h.In the second step, we replace (3.69) with the difference scheme (3.10):
(3.70)ukh(x)-uk-1h(x)+τAhxukh(x)=fk-1h(x)(wtk-wtk-1),1≤k≤N,fk-1h(x)=fh(tk-1,x),tk=kτ,1≤k≤N,x∈Ωh,u0h(x)=∑j=1J‍αju[λj/τ]h(x)+φ(wλ1,…,wλJ,x),x∈Ω~h.Theorem 3.5.
Letτ and |h|=h12+⋯+hn2 be sufficiently small positive numbers. Then, the solution of difference scheme (3.70) satisfies the following convergence estimate:
(3.71)max0≤k≤N(E∥u(tk)-ukh∥L2h2)1/2≤C(δ,λ1)(τ1/2+|h|2),
where C(δ,λ1) do not depend on τ and |h|. Here, one puts u(tk)=u(tk,x)|x∈Ω~h as the grid function of exact solution of problem (3.66) at the grid points t=tk,0≤k≤N and x∈Ω~h.The proof of Theorem3.5 is based on the abstract Theorem 3.3 and the symmetry properties of the difference operator Ahx defined by formula (3.68).


## 4.   Numerical Application

Now, we consider the numerical application of nonlocal boundary value problem:(4.1)dv-vxxdt=e-tsinxdwt,0<t<1,0<x<π,v(0,x,0)=v(1,x,w1)+sinx-e-1sinxw1-e-1sinx,0≤x≤π,v(t,0,wt)=v(t,π,wt)=0,0≤t≤1,wt=tξ,ξ∈N(0,1),0≤t≤1,
for one-dimensional stochastic parabolic equation. For numerical solution of (4.1), we consider the difference scheme 1/2-th order of accuracy in t and second order of accuracy in x for the approximate solution of the nonlocal boundary value problem (4.1):
(4.2)unk-unk-1-un+1k-2unk+un-1kh2τ=f(tk,xn)τ(kτ-(k-1)τ)ξ,f(tk,xn)=e-tksinxn,tk=kτ,xn=nh,1≤k≤N,1≤n≤M-1,u0k=uMk=0,0≤k≤N,un0=unN+sinxn-e-1sinxnw1-e-1sinxn,0≤n≤M.
We will write it in the matrix form
(4.3)Aun+1+Bun+Cun-1=Dφn,1≤n≤M-1,U0=0→,UM=0→.
Here
(4.4)φn=[0φn1·φnN-1φnN](N+1)×1,0→=[00·00](N+1)×1,φnk=f(tk,xn)τ(kτ-(k-1)τ)ξ,1≤k≤N,1≤n≤M-1,A=[00·000a·00·····00·a000·0a](N+1)×(N+1),B=[10·0-1bc·00·····00·c000·bc](N+1)×(N+1),a=(-τh2),b=(-1),c=(1+2τh2),C=A,D=[10·0001·00·····00·1000·00](N+1)×(N+1),us=[us0us1·usN-1usN](N+1)×(1),s=n-1,n,n+1.For the solution of the last matrix equation, we use the modified Gauss elimination method (see [17]). We seek a solution of the matrix equation by the following form:
(4.5)un=αn+1un+1+βn+1,n=M-1,…,1,uM=0→,
where αj, are (N+1)×(N+1) square matrices and βj,  are (N+1)×1 column matrices and (j=1,…,M-1) defined by formulas
(4.6)αn+1=-(B+Cαn)-1A,βn+1=(B+Cαn)-1(Dφn-Cβn),n=1,…,M-1.
Here
(4.7)α1=[00.0000.00.....00.0000.00](N+1)×(N+1),β1=0→.The error between the exact solution and the solutions derived by difference schemes is shown in Table1. To obtain the results we simulated the 1,000 sample paths of Brownian motion for each level of discretization. The estimate (3.71) in Theorem 3.5 suggests that if we double the number of nodes, then the error should be decreased by a factor of 1/2. The theoretical statement for the solution of this difference scheme is supported by the results of the numerical experiment. In fact, we double N and M; the error is even less than half of the previous error.Table 1
Error analysis.
N/M
10/30
20/60
40/120
Difference scheme (4.2)
0.0929
0.0401
0.0187



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*Source: 684248-2012-11-13.xml*