Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise

Abstract

Numerical approximation of a stochastic partial integro-differential equation driven by a space-time white noise is studied by truncating a series representation of the noise, with finite element method for spatial discretization and convolution quadrature for time discretization. Sharp-order convergence of the numerical solutions is proved up to a logarithmic factor. Numerical examples are provided to support the theoretical analysis.

Appendices

Mild solution of (1.1)

In the case $\alpha \in (1,2)$, the boundary condition $\partial _t^{1-\alpha }\psi =0$ is equivalent to $\psi =0$ on $\partial \Omega$ (this can be checked by taking Laplace transform in time). Similarly, in the case $\alpha \in (0,1]$, the boundary condition $\partial _t^{1-\alpha }\psi =0$ is equivalent to $\psi -\psi _0=0$ on $\partial \Omega \times [0,\infty )$, where $\psi _0=\psi (\cdot ,0)$ is the initial value in (1.1).

In the case $\sigma =0$, the solution of the corresponding deterministic problem of (1.1) can be expressed by (via Laplace transform, cf. [23, (3.11) and line 4 of page 12] in the case $\alpha \in (1,2)$)

$$\begin{aligned} \psi (\cdot ,t) = \left\{ \begin{array}{ll} \psi _0 + \int _0^t E(t-s) f(\cdot ,s)\mathrm{d}s &{}\quad \text {if}\,\,\,\alpha \in (0,1] , \\ E(t) \psi _0 + \int _0^t E(t-s) f(\cdot ,s)\mathrm{d}s &{}\quad \text {if}\,\,\,\alpha \in (1,2) , \end{array} \right. \end{aligned}$$

(A.1)

where the operator $E(t):L^2(\mathcal {O})\rightarrow L^2(\mathcal {O})$ is given by

$$\begin{aligned} E(t) \phi :=\frac{1}{2\pi \mathrm{i}}\int _{\varGamma _{\theta ,\kappa }}e^{zt} z^{\alpha -1} (z^\alpha -\Delta )^{-1}\phi \, \mathrm{d}z \quad \forall \, \phi \in L^2(\mathcal {O}) \end{aligned}$$

(A.2)

with integration over a contour $\varGamma _{\theta ,\kappa }$ on the complex plane.

Correspondingly, the mild solution of the stochastic problem (1.1) is defined as (cf. [18, Proposition 2.7] and [24])

$$\begin{aligned} \psi (\cdot ,t) = \left\{ \begin{array}{ll} \psi _0 + \int _0^t E(t-s) f(\cdot ,s)\mathrm{d}s +\sigma \int _0^t E(t-s)\mathrm{d}W(\cdot ,s) &{}\quad \text {if}\quad \alpha \in (0,1] , \\ E(t) \psi _0 + \int _0^t E(t-s) f(\cdot ,s)\mathrm{d}s +\sigma \int _0^t E(t-s)\mathrm{d}W(\cdot ,s) &{}\quad \text {if}\quad \alpha \in (1,2) . \end{array} \right. \end{aligned}$$

(A.3)

For any given initial data $\psi _0\in L^2(\mathcal {O})$ and source $f\in L^1(0,T;L^2(\mathcal {O}))$, the expression (A.3) defines a mild solution $\psi \in C([0,T];L^2(\Omega ; L^2(\mathcal {O})))$. In the case $\psi _0=f=0$ and $\sigma \ne 0$, a simple proof of this result can be found in [12, Appendix]; in the case $\sigma =0$ ($\psi _0$ and f may not be zero), the result is a consequence of the boundedness of the operator $E(t):L^2(\mathcal {O})\rightarrow L^2(\mathcal {O})$, i.e.,

$$\begin{aligned} \Vert E(t)v\Vert&\le C\int _{\varGamma _{\theta ,\kappa }} |e^{zt}| |z|^{\alpha -1} \Vert (z^\alpha -\Delta )^{-1}v\Vert |\mathrm{d}z| \nonumber \\&\le C\Vert v\Vert \int _\kappa ^{\infty } e^{-rt|\cos (\theta )|} r^{-1} \mathrm{d}r +C\Vert v\Vert \int _{-\theta }^\theta e^{\kappa t\cos (\varphi )} \mathrm{d}\varphi \nonumber \\&\le C\Vert v\Vert \qquad \forall \, v\in L^2(\mathcal {O}). \end{aligned}$$

(A.4)

Similarly, the discrete operator $E^{(h)}(t):X_h\rightarrow X_h$ defined by

$$\begin{aligned} E^{(h)}(t) \phi :=\frac{1}{2\pi \mathrm{i}}\int _{\varGamma _{\theta ,\kappa }}e^{zt} z^{\alpha -1} (z^\alpha -\Delta _h )^{-1}\phi \, \mathrm{d}z \quad \forall \, \phi \in X_h , \end{aligned}$$

(A.5)

is also bounded on the finite element subspace $X_h$, i.e.,

$$\begin{aligned} \Vert E^{(h)}(t)v\Vert \le C\Vert v\Vert \quad \forall \, v\in X_h , \end{aligned}$$

(A.6)

where the constant C is independent of the mesh size h.

Representation of the discrete solutions

For $f=0$ we prove the following representation of the solutions of (4.2) and (2.23):

$$\begin{aligned} v^{(h)}(\cdot ,t_n)&= P_h\psi _0 + \frac{1}{2\pi \mathrm {i}}\int _{\varGamma _{\theta ,\kappa }} e^{zt_n}z^{-1} (z^\alpha - \Delta _h)^{-1} \Delta _hP_h\psi _0 \mathrm{d}z , \end{aligned}$$

(B.1)

$$\begin{aligned} v_n^{(h)}&=P_h\psi _0 + \frac{1}{2\pi i}\int _{\varGamma _{\theta ,\kappa }^{(\tau )}} e^{t_nz}e^{-z\tau } \delta (e^{-z\tau })^{-1} \big (\delta (e^{-z\tau })^{\alpha }-\Delta _h\big )^{-1} \Delta _h P_h\psi _0 \, \mathrm{d}z , \end{aligned}$$

(B.2)

which are used in (4.18) in estimating the error of temporal discretization.

In fact, (B.1) is a consequence of (A.3): replacing E(t) by $E^{(h)}(t)$ and substituting $\phi =P_h\psi _0$ yield

$$\begin{aligned} \begin{aligned} v^{(h)}(\cdot ,t_n)&=\frac{1}{2\pi \mathrm{i}}\int _{\varGamma _{\theta ,\kappa }}e^{zt} z^{\alpha -1} (z^\alpha -\Delta _h )^{-1}P_h\psi _0\, \mathrm{d}z \\&=\frac{1}{2\pi \mathrm{i}}\int _{\varGamma _{\theta ,\kappa }}e^{zt} z^{-1}(z^\alpha -\Delta _h+\Delta _h) (z^\alpha -\Delta _h )^{-1}P_h\psi _0\, \mathrm{d}z \\&=\frac{1}{2\pi \mathrm{i}}\int _{\varGamma _{\theta ,\kappa }}e^{zt} z^{-1} P_h\psi _0\, \mathrm{d}z + \frac{1}{2\pi \mathrm{i}}\int _{\varGamma _{\theta ,\kappa }}e^{zt} z^{-1}(z^\alpha -\Delta _h )^{-1}\Delta _hP_h\psi _0\, \mathrm{d}z \\&=P_h\psi _0 + \frac{1}{2\pi \mathrm{i}}\int _{\varGamma _{\theta ,\kappa }}e^{zt} z^{-1}(z^\alpha -\Delta _h )^{-1}\Delta _hP_h\psi _0\, \mathrm{d}z , \end{aligned} \end{aligned}$$

where we have used the identity $\frac{1}{2\pi \mathrm{i}}\int _{\varGamma _{\theta ,\kappa }}e^{zt} z^{-1} \mathrm{d}z =1$ (i.e., the inverse Laplace transform of $z^{-1}$ is 1).

It remains to prove (B.2). To this end, we rewrite (2.23) as

$$\begin{aligned} \bar{\partial }_\tau (v_n^{(h)}-P_h\psi _0) -\Delta _h \bar{\partial }_\tau ^{1-\alpha } (v_n^{(h)}-P_h\psi _0) = \Delta _h \bar{\partial }_\tau ^{1-\alpha } (P_h\psi _0)_n , \end{aligned}$$

(B.3)

where $\bar{\partial }_\tau ^{1-\alpha } (P_h\psi _0)_n:=\frac{1}{\tau ^{1-\alpha }}\sum _{j=1}^n b_{n-j} P_h\psi _0$. Since we are only interested in the solutions $v_n^{(h)}$, $n=1,\dots ,N$, we define

$$\begin{aligned} \widetilde{v}_n^{(h)} =\left\{ \begin{array}{ll} v_n^{(h)} &{}\quad 1\le n\le N,\\ P_h\psi _0 &{}\quad n\ge N+1 , \end{array}\right. \end{aligned}$$

which satisfies the equation

$$\begin{aligned} \bar{\partial }_\tau (\widetilde{v}_n^{(h)}-P_h\psi _0) -\Delta _h \bar{\partial }_\tau ^{1-\alpha } (\widetilde{v}_n^{(h)}-P_h\psi _0) = \Delta _h \bar{\partial }_\tau ^{1-\alpha } (P_h\psi _0)_n + g_n , \end{aligned}$$

(B.4)

with $g_n=0$ for $1\le n\le N$. The right-hand side of (B.4) differs from (B.3) only for $n\ge N+1$, that

$$\begin{aligned} \Vert g_n\Vert\le & {} \Vert \Delta _h \bar{\partial }_\tau ^{1-\alpha } (\widetilde{v}_n^{(h)}-P_h\psi _0)\Vert +\Vert \Delta _h \bar{\partial }_\tau ^{1-\alpha } (P_h\psi _0)_n \Vert \nonumber \\\le & {} \frac{1}{\tau ^{1-\alpha }}\sum _{j=1}^N |b_{n-j}|\Vert \widetilde{v}_j^{(h)}-P_h\psi _0\Vert +\frac{1}{\tau ^{1-\alpha }}\sum _{j=1}^n |b_{n-j}|\Vert P_h\psi _0\Vert \nonumber \\\le & {} C\tau ^{\alpha -1}\left( \sum _{j=1}^N |b_{n-j}|+\sum _{j=1}^n |b_{n-j}|\right) \nonumber \\\le & {} C\tau ^{\alpha -1}n^{\alpha -1}, \end{aligned}$$

as $n\rightarrow \infty$. Thus $\sum _{n=N+1}^\infty g_n\zeta ^n$ is an analytic function of $\zeta$ for $|\zeta |<1$.

By (2.8), summing up (B.4) times $\zeta ^n$ for $n=1,2,\dots$, yields

$$\begin{aligned} \bigg (\frac{1-\zeta }{\tau }-\bigg (\frac{1-\zeta }{\tau }\bigg )^{1-\alpha }\Delta _h \bigg ) \sum _{n=1}^\infty (\widetilde{v}_n^{(h)}-P_h\psi _0)\zeta ^n= \Delta _h \bigg (\frac{1-\zeta }{\tau }\bigg )^{1-\alpha } \frac{\zeta }{1-\zeta } P_h\psi _0 + \sum _{n=N+1}^\infty g_n\zeta ^n , \end{aligned}$$

which implies

$$\begin{aligned} \sum _{n=1}^\infty (\widetilde{v}_n^{(h)}-P_h\psi _0)\zeta ^n= & {} \bigg (\frac{1-\zeta }{\tau }\bigg )^{-1} \bigg (\bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha }-\Delta _h \bigg ) ^{-1}\Delta _h P_h\psi _0 \frac{\zeta }{\tau } \\&+ \bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha -1}\bigg (\bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha }-\Delta _h \bigg )^{-1}\sum _{n=N+1}^\infty g_n\zeta ^n . \end{aligned}$$

For $\kappa >0$ and $\varrho _\kappa =e^{-(\kappa +1)\tau } \in (0,1)$, the Cauchy integral formula implies that

$$\begin{aligned}&\widetilde{v}_n^{(h)}-P_h\psi _0 \nonumber \\&\quad = \frac{1}{2\pi i}\int _{|\zeta |=\varrho _\kappa } \zeta ^{-n-1}\sum _{n=1}^\infty (v_n^{(h)}-P_h\psi _0)\zeta ^n \mathrm{d}\zeta \nonumber \\&\quad = \frac{1}{2\pi i}\int _{|\zeta |=\varrho _\kappa } \zeta ^{-n} \bigg (\frac{1-\zeta }{\tau }\bigg )^{-1} \bigg (\bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha }-\Delta _h\bigg )^{-1}\frac{1}{\tau }\Delta _h P_h\psi _0 \mathrm{d}\zeta \nonumber \\&\quad \quad + \frac{1}{2\pi i}\int _{|\zeta |=\varrho _\kappa } \bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha -1}\bigg (\bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha }-\Delta _h \bigg )^{-1}\sum _{m=N+1}^\infty g_m\zeta ^{m-n-1} \mathrm{d}\zeta . \end{aligned}$$

(B.5)

For $1\le n\le N$ the function $\big (\frac{1-\zeta }{\tau }\big )^{\alpha -1}\big (\big (\frac{1-\zeta }{\tau }\big )^{\alpha }-\Delta _h \big )^{-1}\sum _{m=N+1}^\infty g_m\zeta ^{m-n-1}$ is analytic in $|\zeta |<1$. Consequently, Cauchy’s integral theorem implies

$$\begin{aligned} \frac{1}{2\pi i}\int _{|\zeta |=\varrho _\kappa } \bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha -1}\bigg (\bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha }-\Delta _h \bigg )^{-1}\sum _{m=N+1}^\infty g_m\zeta ^{m-n-1} \mathrm{d}\zeta =0 . \end{aligned}$$

Substituting this identity into (B.5) yields, for $1\le n\le N$,

$$\begin{aligned}&\widetilde{v}_n^{(h)}-P_h\psi _0\nonumber \\&\quad =v_n^{(h)}-P_h\psi _0 \nonumber \\&\quad = \frac{1}{2\pi i}\int _{|\zeta |=\varrho _\kappa } \zeta ^{-n} \bigg (\frac{1-\zeta }{\tau }\bigg )^{-1} \bigg (\bigg (\frac{1-\zeta }{\tau }\bigg )^{\alpha }-\Delta _h\bigg )^{-1}\frac{1}{\tau }\Delta _h P_h\psi _0 \mathrm{d}\zeta \nonumber \\&\quad = \frac{1}{2\pi i}\int _{\varGamma ^\tau }e^{t_nz}e^{-z\tau } \bigg (\frac{1-e^{-\tau z}}{\tau }\bigg )^{-1} \bigg (\bigg (\frac{1-e^{-\tau z}}{\tau }\bigg )^{\alpha }-\Delta _h\bigg )^{-1} \Delta _h P_h\psi _0 \, \mathrm{d}z \nonumber \\&= \frac{1}{2\pi i}\int _{\varGamma ^\tau }e^{t_nz}e^{-z\tau } \delta (e^{-z\tau })^{-1} \big (\delta (e^{-z\tau })^{\alpha }-\Delta _h\big )^{-1} \Delta _h P_h\psi _0 \, \mathrm{d}z, \end{aligned}$$

(B.6)

where we have used the change of variable $\zeta =e^{-z\tau }$, which converts the path of integration to the contour

$$\begin{aligned} \varGamma ^\tau =\left\{ z=\kappa +1+\mathrm {i} y: \, y\in {\mathbb R}\quad \text {and}\quad |y|\le {\pi }/{\tau } \right\} . \end{aligned}$$

The angle condition (3.4) and [3, Theorem 3.7.11] imply that the integrand on the right-hand side of (B.6) is analytic in the region

$$\begin{aligned} \Sigma _{\theta ,\kappa }^\tau = \Big \{z\in {\mathbb C} : |\mathrm{arg}(z)|\le \theta ,\quad |z|\ge \kappa ,\quad |\mathrm{Im}(z)|\le \frac{\pi }{\tau },\quad \mathrm{Re}(z)\le \kappa +1 \Big \} , \end{aligned}$$

enclosed by the four paths $\varGamma ^\tau$, $\varGamma _{\theta ,\kappa }^{(\tau )}$ and ${\mathbb R}\pm \mathrm {i}\pi /\tau$, where $\varGamma _{\theta ,\kappa }^{(\tau )} \, =\left\{ z\in \varGamma _{\theta ,\kappa } : |\mathrm{Im}(z)|\right. \left. \le \frac{\pi }{\tau }\right\}$. Then Cauchy’s theorem allows us to deform the integration path from $\varGamma ^\tau$ to $\varGamma _{\theta ,\kappa }^{(\tau )}$ in the integral (B.6) (the integrals on ${\mathbb R}\pm \mathrm {i}\pi /\tau$ cancels each other). This yields the desired representation (B.2).

Some inequalities

In this appendix, we prove the following two inequalities:

$$\begin{aligned}&C_0^{\#}|z|\tau \le |1-e^{z\tau }|\le C_1^{\#}|z|\tau ,&\forall \, z\in \varGamma _{\theta ,\kappa }^{(\tau )}, \end{aligned}$$

(C.1)

$$\begin{aligned}&|1-e^{z\tau }|\le C|z|^{1/q}\tau ^{1/q},&\forall \, z\in \varGamma _{\theta ,\kappa },\quad 1\le q\le \infty , \end{aligned}$$

(C.2)

which have been used in (3.27), (4.26) and (4.32).

Proof of (C.1)

Note that

$$\begin{aligned} \varGamma _{\theta ,\kappa }^{(\tau )}&=\left\{ z\in \mathbb {C}: |z|=\kappa ,\, |\arg z|\le \theta \right\} \cup \left\{ z\in \mathbb {C}: z=\rho e^{\pm \mathrm{i}\theta }, \rho \ge \kappa , |\mathrm{Im}(z)|\le \frac{\pi }{\tau } \right\} \nonumber \\&=: \varGamma _{\theta ,\kappa }^{(\tau ),1}\cup \varGamma _{\theta ,\kappa }^{(\tau ),2}. \end{aligned}$$

(C.3)

For $z\in \varGamma _{\theta ,\kappa }^{(\tau )}$ we have $|z|\tau \le \pi /\sin (\theta )$. Since $|z|\tau$ is bounded, the following Taylor expansion holds:

$$\begin{aligned}&1-e^{z\tau } =-z\tau +O(|z|^2\tau ^2) , \end{aligned}$$

(C.4)

which implies

$$\begin{aligned} |1-e^{z\tau }| \le C_1^{+}|z|\tau ,\quad \text {if}\,\,\,z\in \varGamma _{\theta ,\kappa }^{(\tau )}. \end{aligned}$$

This proves the right-half inequality of (C.1).

From (C.4) we also see that there exists a small constant $\gamma$ such that

$$\begin{aligned} C_0^{+}|z|\tau \le |1-e^{z\tau }| ,\quad \text {if}\,\,\,z\in \varGamma _{\theta ,\kappa }^{(\tau )},\,\,\, |z|\tau <\gamma . \end{aligned}$$

(C.5)

If $|z|\tau \ge \gamma$, then the following inequality holds for $\theta$ satisfying the condition of Lemma 1:

$$\begin{aligned} \gamma \le |z|\tau \le \frac{\pi }{\sin (\theta )}\le \pi \sqrt{1+4/\pi ^2} \le \frac{3}{2}\pi . \end{aligned}$$

Since the function $g(w):=|1-e^{w}|$ is not zero for $\gamma \le |w| \le \frac{3}{2}\pi$, the function g(w) must have a positive minimum value $\varpi$ for $\gamma \le |w| \le \frac{3}{2}\pi$, i.e., $g(w)\ge \varpi$. Consequently, we have

$$\begin{aligned} \begin{aligned}&\varpi \frac{\sin (\theta )}{\pi }|z|\tau \le \varpi \le |1-e^{z\tau }| ,&\text {if}\,\,\,z\in \varGamma _{\theta ,\kappa }^{(\tau )} ,\,\,\, |z|\tau \ge \gamma , \end{aligned} \end{aligned}$$

(C.6)

where we have used the inequality $\frac{\sin (\theta )}{\pi }|z|\tau \le 1$ in the last inequality. Combining (C.5) and (C.6) yields (C.1).$\square$

Proof of (C.2)

If $z\in \varGamma _{\theta ,\kappa }$ and $|z|\tau \le \pi /\sin (\theta )$, then $z\in \varGamma _{\theta ,\kappa }^{(\tau )}$. In this case, (C.1) implies

$$\begin{aligned}&|1-e^{z\tau }|\le C|z|\tau&\forall \, z\in \varGamma _{\theta ,\kappa } , \,\,\, |z|\tau \le \pi /\sin (\theta ),\\&|1-e^{z\tau }|\le C&\forall \, z\in \varGamma _{\theta ,\kappa } , \,\,\, |z|\tau \le \pi /\sin (\theta ) . \end{aligned}$$

The combination of the two inequalities above yields

$$\begin{aligned}&|1-e^{z\tau }|\le C|z|^{1/q}\tau ^{1/q}&\forall \, z\in \varGamma _{\theta ,\kappa } , \,\,\, |z|\tau \le \pi /\sin (\theta ). \end{aligned}$$

(C.7)

If $z\in \varGamma _{\theta ,\kappa }$ and $|z|\tau \ge \pi /\sin (\theta )$, then

$$\begin{aligned} |e^{z\tau }|=e^{-|z|\tau \cos (\theta )}\le e^{-\pi /\tan (\theta )}, \end{aligned}$$

which implies

$$\begin{aligned}&|1-e^{z\tau }|\le 1+e^{-\pi /\tan (\theta )} \le 2 \le 2 \bigg (\frac{\sin (\theta )}{\pi }\bigg )^{\frac{1}{q}} |z|^{1/q}\tau ^{1/q}&\forall \, z\in \varGamma _{\theta ,\kappa } , \quad |z|\tau \ge \pi /\sin (\theta ). \end{aligned}$$

(C.8)

Combining (C.7) and (C.8) yields (C.2).$\square$