Convergence of tamed Euler schemes for a class of stochastic evolution equations

Abstract

We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done by using the recently developed tamed Euler method, which employs a fully explicit time stepping, coupled with a Galerkin scheme for the spatial discretization.

1 Introduction

In this paper we investigate the convergence of full discretizations, explicit in time, of stochastic evolution equations

$$\begin{aligned} du(t) = A u(t) dt + B u(t) dW(t)\, , \,\, t \in [0,T] \end{aligned}$$

(1.1)

with the drift term governed by a super-linearly growing operator. When the operator appearing in the drift term grows at most linearly then the classical explicit Euler scheme applied to stochastic evolution equations is convergent (when coupled appropriately with the spatial discretization), see, for example, Gyöngy and Millet [6]. If the operator appearing in the drift term grows faster than linearly then one would, in general, not expect the explicit Euler scheme to be convergent (this is the case even in the setting of fully deterministic evolution equations). Instead, one would typically consider the implicit Euler scheme which is convergent in this situation (see, for example, Gyöngy and Millet [6]). The price one pays is the increased computational effort required at each time step of the numerical scheme.

Hutzenthaler, Jentzen and Kloeden [9] have observed that the absolute moments of explicit Euler approximations for stochastic differential equations with super-linearly growing coefficients may diverge to infinity at finite time. This led to development of “tamed” Euler schemes for stochastic differential equations. This was pioneered in Hutzenthaler, Jentzen and Kloeden [10] and, using different techniques, in Sabanis [16]. A taming-like technique in the form of truncation has been proposed by Roberts and Tweedie [14] in the context of Markov chain Monte Carlo methods. Further work on explicit numerical approximations of stochastic differential equations with super-linearly growing coefficients can be found in Tretyakov and Zhang [19], Hutzenthaler and Jentzen [12], Sabanis [17] as well Dareiotis, Kumar and Sabanis [18].

Moreover Hutzenthaler, Jentzen and Kloeden [11] have demonstrated that to apply multilevel Monte Carlo methods (see Heinrich [7, 8] and Giles [5]) to stochastic differential equations with super-linearly growing coefficients one must “tame” the explicit Euler scheme. In this paper we use the idea of “taming” to devise a new type of a convergent explicit scheme for a class of stochastic evolution equations with super-linearly growing operators in the drift term.

The article is organised as follows. In Sect. 2 we introduce the numerical scheme, give the precise assumptions and state the main result in Theorem 2. Essential a priori estimates are proved in Sect. 3. In Sect. 4 we first use the a priori estimates and a compactness argument to extract weakly convergent subsequences and limits of the approximation. The remaining step is to identify the weak limit of the approximation of the nonlinear term with the nonlinear term in the equation. This is done using a monotonicity argument in Sect. 4 where Theorem 2 is finally proved. In Sect. 5 we provide examples of stochastic partial differential equations where the numerical scheme can be applied.

2 Main results

Let $T>0$. Let $(\varOmega , \mathcal {F}, \mathbb {P})$ be a probability space and let $(\mathcal {F}_t)_{t\in [0,T]}$ be a filtration such that $\mathcal {F}_0$ contains all the $\mathbb {P}$-null sets of $\mathcal {F}$.

Let $K>0$ and $p \in [2,\infty )$ be given constants. Let $p^* := p / (p-1)$. For a reflexive, separable Banach space $(X,\Vert \cdot \Vert _X)$ let $X^*$ and $\Vert \cdot \Vert _{X^*}$ denote its dual space and the norm on the dual space respectively. For $f\in X^*$ and $v\in X$ we use $\langle f,v \rangle$ to denote the duality pairing. By $L^p(0,T;X)$ we denote the Lebesgue–Bochner space of equivalence classes of measurable functions with values in X that satisfy

$$\begin{aligned} \Vert x\Vert _{L^p(0,T;X)} := \left( \int _0^T\Vert x(t)\Vert _X^p dt\right) ^{1/p} < \infty . \end{aligned}$$

By $L^p(\varOmega ; X)$ we denote the Lebesgue–Bochner space of random variables with values in X and such that the norm

$$\begin{aligned} \Vert x\Vert _{L^p(\varOmega ;X)} := \left( \mathbb {E}(\Vert x\Vert _X^p)\right) ^{1/p} \end{aligned}$$

is finite. Finally by $\mathcal {L}^p(X)$ we denote the Lebesgue–Bochner space of $dt\times \mathbb {P}$-equivalence classes of $(\mathcal {F}_t)_{t\in [0,T]}$-adapted and X-valued stochastic process that satisfy

$$\begin{aligned} \Vert x\Vert _{\mathcal {L}^p(X)} := \left( \mathbb {E}\int _0^T \Vert x(t)\Vert _X^p dt \right) ^{1/p} < \infty . \end{aligned}$$

We say that an operator $C:X\times \varOmega \rightarrow X^*$ is measurable with respect to some $\mathcal {G} \subseteq \mathcal {F}$ if for any $v,w\in X$ the real-valued random variable $\langle C v,w\rangle$ is $\mathcal {G}$-measurable.

We assume that, with respect to $(\mathcal {F}_t)_{t\in [0,T]}$, $(W_t)_{t\in [0,T]}$ is a cylindrical Q-Wiener process with $Q=I$ on a separable Hilbert space $(U,(\cdot ,\cdot )_U, |\cdot |_U)$. We assume that there are $(V_1,\Vert \cdot \Vert _{V_1})$ and $(V_2,\Vert \cdot \Vert _{V_2})$, separable and reflexive Banach spaces that are densely and continuously embedded in H, where $(H,(\cdot ,\cdot ),|\cdot |)$ is a Hilbert space identified with its dual. We thus have two Gelfand triples

$$\begin{aligned} V_i \hookrightarrow H \hookrightarrow V_i^*\,, \,\,\, i \in \{1,2\}. \end{aligned}$$

Let $A_i$ with $i\in \{1,2\}$ be operators defined on $V_i\times \varOmega$ with values in $V_i^*$. Let $B_i$ with $i\in \{1,2\}$ be operators defined on $V_i\times \varOmega$ with values in $L_2(U,H)$, where $L_2(U,H)$ is the space of Hilbert–Schmidt operators from U to H.

Let $V:=V_1 \cap V_2$ and let the norm in V be given by $\Vert \cdot \Vert := \Vert \cdot \Vert _{V_1} + \Vert \cdot \Vert _{V_2}$. Assume that V is separable and dense in both $V_1$ and $V_2$. Using Gajewski, Gröger and Zacharias [4, Kapitel I,Satz 5.13] one observes that the dual $V^*$ of V can be identified with

$$\begin{aligned} V_1^* + V_2^* := \{f = f_1 + f_2 : f_1 \in V_1^*,\, f_2\in V_2^*\} \end{aligned}$$

and that for all $f \in V^*$

$$\begin{aligned} \Vert f\Vert _{V*} = \inf \{\max (\Vert f_1\Vert _{V_1^*}, \Vert f_2\Vert _{V_2^*}): f = f_1 + f_2, f_1 \in V_1^*,\,\, f_2\in V_2^* \}. \end{aligned}$$

We consider stochastic evolution equations of the form

$$\begin{aligned} du(t) = \big [A_1 u(t) + A_2 u(t)\big ] dt + \big [B_1 u(t) + B_2 u(t)\big ] dW(t)\, , \,\, t \in [0,T], \end{aligned}$$

(2.1)

where $u(0) = u_0$ with $u_0$ a given H-valued and $\mathcal {F}_0$-measurable random variable. Let $A:=A_1 + A_2$ and $B:=B_1 + B_2$. The operator A is defined on $V\times \varOmega$ with values in $V^*$ and the operator B is defined on $V\times \varOmega$ with values in $L_2(U,H)$. Then we can write (2.1) as (1.1).

We impose the following assumptions on the operators.

Assumption 1

Let $A_i : V_i \times \varOmega \rightarrow V_i^*$ be $\mathcal {F}_0$-measurable operators for $i\in \{1,2\}$. Let $B_i: V_i \times \varOmega \rightarrow L_2(U,H)$ be such that for any $v\in V_i$, $u\in U$ and $h\in H$ the real-valued random variable $((B_i v)u,h)$ is $\mathcal {F}_0$-measurable for $i\in \{1,2\}$. Moreover assume that the following conditions hold.

Monotonicity:

$$\begin{aligned} 2\langle A v - A w, v-w\rangle + \Vert B v-B w\Vert _{L_2(U,H)}^2 \le K|v-w|^2\,\, \text { for all }\,\, v,w \in V. \end{aligned}$$

Coercivity: there is $\mu > 0$ such that

$$\begin{aligned} 2\langle A_1 v, v \rangle + \Vert B_1 v\Vert _{L_2(U,H)}^2 \le - \mu \Vert v\Vert _{V_1}^2 + K(1+|v|^2)\,\, \text { for all }\,\, v\in V_1. \end{aligned}$$

and

$$\begin{aligned} 2\langle A_2 v, v \rangle + \Vert B_2 v\Vert _{L_2(U,H)}^2 \le K(1+|v|^2)\,\, \text { for all }\,\, v\in V_2. \end{aligned}$$

Growth:

$$\begin{aligned} \Vert A_1 v\Vert _{V_1^*}^2 \le K(1+\Vert v\Vert _{V_1}^2)\,\, \text { for all }\,\, v\in V_1 \end{aligned}$$

and

$$\begin{aligned} \Vert A_2 v\Vert _{V_2^*}^{p^*} \le K(1+\Vert v\Vert _{V_2}^p)\,\, \text { for all }\,\, v\in V_2. \end{aligned}$$

as well as

$$\begin{aligned} \Vert Bv\Vert _{L_2(U,H)}^2 \le K(1+|v|^2) \,\, \text { for all }\,\, v\in H. \end{aligned}$$

Hemicontinuity: for any v, w and z in V

$$\begin{aligned} \langle A(v+\epsilon w),z\rangle \rightarrow \langle Av,z \rangle \,\,\text { as }\,\, \epsilon \rightarrow 0. \end{aligned}$$

We now define what is meant by solution of (1.1).

Definition 1

(Solution) Let $u_0$ be an $\mathcal {F}_0$-measurable H-valued random variable. We say that a continuous, H-valued and $(\mathcal {F}_t)_{t\in [0,T]}$-adapted process u is a solution to (1.1) if u is $dt \times \mathbb {P}$ almost everywhere V-valued, if $u\in \mathcal {L}^2(V_1)\cap \mathcal {L}^p(V_2)$ and if for every $t\in [0,T]$ and every $v\in V$, almost surely,

$$\begin{aligned} ( u(t),v ) = ( u_0, v ) + \int _0^t \langle A u(s), v \rangle ds + \int _0^t (v,Bu(s) dW(s)). \end{aligned}$$

To the best knowledge of the authors, existence and uniqueness has not been proved for this class of stochastic evolution equations. Pardoux [13] considers the situation where the stochastic evolution equation is governed by a sum of monotone, coercive and hemicontinuous operators satisfying certain growth condition. However the operator $A_2$ in our case only satisfies a type of “degenerate” coercivity condition. Hence the existence theorem from Pardoux [13] does not apply. We prove that a solution to (1.1) must be unique in Theorem 1 and we prove existence of the solution in Theorem 2.

Theorem 1

(Uniqueness) The solution of (1.1), specified by Definition 1, is unique, provided that the Growth and Monotonicity conditions in Assumption 1 are satisfied.

We prove Theorem 1 in Sect. 4. Let us now describe the discretization scheme for the stochastic evolution Eq. (2.1). For the space discretization let $(V_m)_{m \in \mathbb {N}}$ be a Galerkin scheme for V. To be precise we assume that $V_m\subseteq V$ are finite dimensional spaces with the dimension of $V_m$ equal to m. We further assume that $V_m \subseteq V_{m+1}$ for all $m\in \mathbb {N}$ and that

$$\begin{aligned} \lim _{m\rightarrow \infty } \inf \{\Vert v-\varphi \Vert \, :\, \varphi \in V_m\}=0\,\,\,\, \forall v\in V. \end{aligned}$$

(this is referred to as the limited completeness of the Galerkin scheme). We will need the following projection operators.

Assumption 2

For any $m\in \mathbb {N}$ let $\varPi _m:V^* \rightarrow V_m$ satisfy the following:

• 1. For any $v \in V_m$, $\varPi _m v = v$.

• 2. If $f\in V^*$ and $v\in V$ then $\langle f, \varPi _m v \rangle = \langle v, \varPi _m f \rangle$.

• 3. If $g,h \in H$ then $(\varPi _m g,h) = (\varPi _m h, g)$ and $|\varPi _m h| \le |h|$.

• 4. There is a constant, depending on m and denoted by $\mathfrak {c}(m)$, such that

  $$\begin{aligned} |\varPi _m f|^2 \le \mathfrak {c}(m)\Vert f\Vert _{V^*}^2\,\, \text { for all }\,\, f\in V^*. \end{aligned}$$

In applications this assumption is easily satisfied. In particular if $\{\varphi _j \in V\,\, : \,\, j = 1,2,\ldots \}$ is an orthonormal basis in H then taking $V_m := \text {span}\{\varphi _1,\ldots ,\varphi _m\}$ is a Galerkin scheme for V. Taking $\varPi _m f := \sum _{j=1}^m \langle f, \varphi _j \rangle \varphi _j$ satisfies the first three conditions in Assumption 2. Moreover, the following holds

$$\begin{aligned} |\varPi _m f|^2 = \bigg | \sum _{j=1}^m \langle f, \varphi _j\rangle \varphi _j \bigg |^2 = \sum _{j=1}^m \langle f, \varphi _j\rangle ^2 \le \Vert f\Vert _{V^*}^2 \sum _{j=1}^m \Vert \varphi _j\Vert _V^2 = \mathfrak {c}(m) \Vert f\Vert _{V^*}^2, \end{aligned}$$

where $\mathfrak {c}(m) := \sum _{j=1}^m \Vert \varphi _j\Vert _V^2$. Thus the fourth condition in Assumption 2 is also satisfied. Let $\{\chi _i\}_{i\in \mathbb {N}}$ be an orthonormal basis of U. Fix $k \in \mathbb {N}$ and define

$$\begin{aligned} W_k(t) := \sum _{j=1}^{k} (W(t),\chi _j)_U \chi _j. \end{aligned}$$

For the time discretization take $n\in \mathbb {N}$, let $\tau _n := T/n$ and define the grid points on an equidistant grid as $t^n_i := \tau _n i$, $i=0,1,\ldots ,n$. Further consider some sequence $((n_\ell , m_\ell , k_\ell ))_{\ell \in \mathbb {N}}$ such that $n_\ell , m_\ell$ and $k_\ell$ all go to infinity as $\ell \rightarrow \infty$.

Let c denote a generic positive constant that may depend on T, on the constants arising in the continuous embeddings $V_i \hookrightarrow H \hookrightarrow V_i^*$, $i=1,2$ and on the constants arising in Assumptions 1 and 3 but that is always independent of the discretization parameters m, k and n. Define $\kappa _{n_\ell }(t) = t^{n_\ell }_i$ if $t\in [t^{n_\ell }_i, t^{n_\ell }_{i+1})$ for $i=0,\ldots ,n_\ell -1$ and $\kappa _{n_\ell }(T) = T$. Fix some $\ell \in \mathbb {N}$ (and hence $m_\ell$, $n_\ell$ and $k_\ell$). Let $u_\ell (0)$ be a $V_{m_\ell }$ valued $\mathcal {F}_0$-measurable approximation of $u_0$. For example we can take $u_\ell (0) := \varPi _{m_\ell } u_0$ but other approximations are possible. For $t>0$ we define a process $u_\ell$ by

$$\begin{aligned} u_\ell (t)= & {} u_\ell (0) + \int _0^t \varPi _{m_\ell } \left[ A_1 u_\ell (\kappa _{n_\ell }(s)) + A_{2,\ell } u_\ell (\kappa _{n_\ell }(s)) \right] ds\nonumber \\&+ \int _0^t \varPi _{m_\ell } B u_\ell (\kappa _{n_\ell }(s))dW_{k_\ell }(s), \end{aligned}$$

(2.2)

where we use the “tamed” operator $A_{2,\ell }$ defined by

$$\begin{aligned} A_{2,\ell } v := \frac{1}{1+{n_\ell }^{-1/2}|\varPi _{m_\ell } A_2 v|}A_2 v \end{aligned}$$

(2.3)

for any $v \in V_2$. We will use the following notation: $\bar{u}_\ell (t) := u_\ell (\kappa _{n_\ell }(t))$ and $a_\ell (v) := \varPi _{m_\ell }[A_1 v + A_{2,\ell } v]$. Then (2.2) is equivalent to

$$\begin{aligned} u_\ell (t) = u_\ell (0) + \int _0^t a_\ell (\bar{u}_\ell (s)) ds + \int _0^t \varPi _{m_\ell } B \bar{u}_\ell (s) dW_{k_\ell }(s). \end{aligned}$$

(2.4)

In particular at the time-grid points we have, for $i=0,1,\ldots ,n_\ell -1$,

$$\begin{aligned} u_\ell (t^{n_\ell }_{i+1}) = u_\ell (t^{n_\ell }_i) + a_\ell (u_\ell (t^{n_\ell }_i))\tau _{n_\ell } + \varPi _{m_\ell } B u_\ell (t^{n_\ell }_i) \varDelta W_{k_\ell }(t_{i+1}), \end{aligned}$$

where $\varDelta W_{k_\ell }(t^{n_\ell }_{i+1}) := W_{k_\ell }(t^{n_\ell }_{i+1})-W_{k_\ell }(t^{n_\ell }_i)$. This in turn is equivalent to

$$\begin{aligned} \frac{u_\ell (t^{n_\ell }_{i+1}) - u_\ell (t^{n_\ell }_i)}{\tau _{n_\ell }} = a_\ell (u_\ell (t^{n_\ell }_i)) + \varPi _{m_\ell } Bu_\ell (t^{n_\ell }_i) \frac{\varDelta W_{k_\ell }(t^{n_\ell }_{i+1})}{\tau _{n_\ell }}. \end{aligned}$$

We list below the properies which are satisfied by the tamed operator $A_{2,\ell }$. These are consequences of its structure and the assumed properties of $A_2$. For brevity let, for any $v\in V_2$,

$$\begin{aligned} T_\ell (v) := \frac{1}{1+n_\ell ^{-1/2}|\varPi _{m_\ell } A_2 v|}. \end{aligned}$$

(2.5)

Then for any $v\in V_2$,

$$\begin{aligned} |\varPi _{m_\ell } A_{2,\ell } v| = T_\ell (v)|\varPi _{m_\ell } A_2 v| \le n_\ell ^{1/2} \end{aligned}$$

(2.6)

and also, using the Growth assumption on $A_2$,

$$\begin{aligned} \Vert A_{2,\ell } v\Vert _{V_2^*}^{p^*} = T_\ell (v)^{p^*}\Vert A_2 v\Vert _{V_2^*}^{p^*} \le K(1+\Vert v\Vert _{V_2}^p). \end{aligned}$$

(2.7)

Furthermore, using the Coercivity assumption on $A_2$, we note that for all $v \in V_{m_\ell }$ we have

$$\begin{aligned} 2\langle A_{2,\ell } v, v \rangle = 2T_\ell (v)\langle A_2 v,v \rangle \le K(1+|v|^2). \end{aligned}$$

(2.8)

Thus the weaker coercivity assumption that has been made about $A_2$ is retained. Consider, for a moment, that $A_2$ satisfies the “usual” coercivity condition

$$\begin{aligned} 2\langle A_2 v, v \rangle + \Vert B_2 v\Vert _{L_2(U,H)}^2 \le - \mu \Vert v\Vert _{V_2}^p + K(1+|v|^2)\,\, \text { for all }\,\, v\in V_2. \end{aligned}$$

We see that in this case the best coercivity we can get from this for $A_{2,\ell }$ is again only (2.8). Hence to obtain the necessary a priori estimates we will need an interpolation inequality between $V_2$ and $V_1$ with H.

Assumption 3

There are constants $\lambda \in [0,2/p)$ and $\varLambda >0$ such that for any $v\in V$

$$\begin{aligned} \Vert v\Vert _{V_2} \le \varLambda \Vert v\Vert _{V_1}^\lambda |v|^{1-\lambda }. \end{aligned}$$

Note that in order to overcome the difficulty with coercivity it would suffice to have Assumption 3 satisfied with $\lambda \in [0,2/p]$. However monotonicity of $A_2$ is not preserved by taming. To overcome this we will need to show that $A_{2,\ell } \bar{u}_\ell - A_2 \bar{u}_\ell \rightarrow 0$ in $\mathcal {L}^{p*}(V_2^*)$. To achieve this we use the fact that $\lambda \in [0,2/p)$ in Lemma 4 and the following observation: Assumption 3 implies that there is $\eta > 0$ such that

$$\begin{aligned} \Vert v\Vert _{V_2}^{p(1+\eta )} \le c \Vert v\Vert _{V_1}^2 |v|^\rho , \end{aligned}$$

where $\rho := (1-\lambda )p(1+\eta )$. From this it follows that if $v\in L^2(\varOmega ; L^2(0,T;V_1))$ and $v\in L^{2\rho }(\varOmega ; L^\infty (0,T;H))$ then

$$\begin{aligned} \mathbb {E}\int _0^T \Vert v(s)\Vert _{V_2}^{p(1+\eta )} ds \le c\Bigg [ \mathbb {E}\sup _{s\le t}|v(s)|^{2\rho } + \mathbb {E}\bigg (\int _0^T \Vert v(s)\Vert _{V_1}^2 ds\bigg )^2 \Bigg ]. \end{aligned}$$

Thus we see that Assumption 3 allows us to control the approximate solution in the $L^{p(1+\eta )}((0,T)\times \varOmega ; V_2)$ norm, provided that we can control the approximate solution in the norms of $L^2(\varOmega ; L^2(0,T;V_1))$ and $L^{2\rho }(\varOmega ; L^\infty (0,T;H))$.

Let us take $q_0 := \max (4,2\rho )$. Now we can state the main result of this paper.

Theorem 2

Let Assumptions 1, 2 and 3 be satisfied. Let $u_0 \in L^{q_0}(\varOmega ; H)$ and let $u_\ell (0) \rightarrow u_0$ in $L^{q_0}(\varOmega ; H)$. Assume that $\tfrac{\mathfrak {c}(m_\ell )}{n_{\ell }} \rightarrow 0$ as $\ell \rightarrow \infty$. Then there exists a unique solution u to (1.1) and $\bar{u}_\ell \rightharpoonup u$ in $\mathcal {L}^2(V_1)$ and in $\mathcal {L}^p(V_2)$ and $u_\ell (T) \rightarrow u(T)$ in $L^2(\varOmega ;H)$ as $\ell \rightarrow \infty$.

In Sect. 5 we provide examples of stochastic partial differential equations where Theorem 2 can be applied. We also compute $\mathfrak {c}(m)$ in case of the spectral Galerkin method to make the implications of the space-time coupling constraint more explicit. The crucial point is that the requirement is no more onerous than in the case of equations with operators growing at most linearly.

3 A priori estimates

We start with an important observation that allows us to use standard results on bounds of stochastic integrals driven by finite dimensional Wiener processes.

Remark 1

Recall that $(\chi _j)_{j\in \mathbb {N}}$ is an orthonormal basis in U. Moreover recall that $\bar{u}_\ell (t) := u_\ell (\kappa _{n_\ell }(t))$ and that $a_\ell (v) := \varPi _{m_\ell }[A_1 v + A_{2,\ell } v]$. For each $j\in \mathbb {N}$ a Wiener processes $\mathcal {W}_j$ is obtained by taking $\mathcal {W}_j(t) := (W(t),\chi _j)_U$. If $i\ne j$ then $\mathcal {W}_i$ and $\mathcal {W}_j$ are independent. Furthermore (2.4) is equivalent to

$$\begin{aligned} u_\ell (t) = u_\ell (0) + \int _0^t a_\ell (\bar{u}_\ell (s))ds + \sum _{j=1}^{k_\ell } \int _0^t \varPi _{m_\ell } B \bar{u}_\ell (s) \chi _j d\mathcal {W}_j(s). \end{aligned}$$

Fix $\ell \in \mathbb {N}$ (and thus $k_\ell , m_\ell$ and $n_\ell$ are also fixed). Then using the Growth assumptions on $A_1$ and B, Assumption 2 as well as (2.6) one observes that $|a_\ell (v)|^2 \le 2\mathfrak {c}(m)K(1+|v|^2) + 2n_\ell$ and $|\varPi _{m_\ell }B v\chi _j|^2 \le K(1+|v|^2)$. Hence one knows that, for $q\ge 1$,

$$\begin{aligned} \mathbb {E}\sup _{0\le t \le T} |u_\ell (t)|^q < \infty , \end{aligned}$$

provided that $\mathbb {E}|u_\ell (0)|^q < \infty$. Clearly, at this point, one cannot claim that this bound is independent of $\ell$.

One applies Itô’s formula to (2.4) to obtain

$$\begin{aligned} |u_\ell (t)|^2= & {} |u_\ell (0)|^2 + \int _0^t \bigg [2 \langle a_\ell (\bar{u}_\ell (s)),u_\ell (s)\rangle + \sum _{j=1}^{k_\ell }|\varPi _{m_\ell } B \bar{u}_\ell (s)\chi _j|^2\bigg ] ds\\&+ \int _0^t 2 (B \bar{u}_\ell (s), u_\ell (s) dW_{k_\ell }(s))\,, \end{aligned}$$

which can be rewritten as

$$\begin{aligned} |u_\ell (t)|^2= & {} |u_\ell (0)|^2 + \int _0^t \bigg [2 \langle a_\ell (\bar{u}_\ell (s)),\bar{u}_\ell (s)\rangle + \sum _{j=1}^{k_\ell }|\varPi _{m_\ell } B \bar{u}_\ell (s)\chi _j|^2 \bigg ] ds\nonumber \\&+\, 2\int _0^t \langle a_\ell (\bar{u}_\ell (s)),u_\ell (s)-\bar{u}_\ell (s)\rangle ds\nonumber \\&+\, \int _0^t 2 (B \bar{u}_\ell (s), u_\ell (s) dW_{k_\ell }(s))\, , \end{aligned}$$

(3.1)

in order to apply the coercivity assumption so as to obtain the a priori estimates for the discretized equation.

First we concentrate on the term that arises from the “correction” that one has to make to use the coercivity assumption due to the use of an explicit scheme.

Lemma 1

Let the Growth condition in Assumption 1 be satisfied. Let Assumption 2 hold. Let $q\ge 1$ be given. Then

$$\begin{aligned}&\mathbb {E}\bigg ( \frac{1}{\tau _{n_\ell }} \int _0^t |u_\ell (s)-\bar{u}_\ell (s)|^2 ds \bigg )^q\nonumber \\&\quad \le c_{T,q} \Bigg (1+ (\mathfrak {c}(m)\tau _{n_\ell })^q \mathbb {E}\bigg (\int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^q + \mathbb {E}\int _0^t |\bar{u}_\ell (s)|^{2q}ds \Bigg ) \end{aligned}$$

(3.2)

and

$$\begin{aligned}&\mathbb {E}\bigg ( \int _0^t |a_\ell (\bar{u}_\ell (s))||u_\ell (s)-\bar{u}_\ell (s)|ds\bigg )^q \nonumber \\&\le c_{T,q}\Bigg (1+ (\mathfrak {c}(m)\tau _{n_\ell })^q \mathbb {E}\bigg (\int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^q + \mathbb {E}\int _0^t |\bar{u}_\ell (s)|^{2q}ds \Bigg ). \end{aligned}$$

(3.3)

Proof

From (2.2) it is clear that

$$\begin{aligned} I_{1,\ell }(t):= & {} \mathbb {E}\bigg ( \frac{1}{\tau _{n_\ell }} \int _0^t |u_\ell (s)-\bar{u}_\ell (s)|^2 ds \bigg )^q \\= & {} \mathbb {E}\bigg ( \int _0^t \frac{1}{\tau _{n_\ell }}\bigg | \int _{\kappa _{n_\ell }(s)}^s a_\ell (\bar{u}_\ell (r)) dr + \int _{\kappa _{n_\ell }(s)}^s \varPi _{m_\ell } B \bar{u}_\ell (r) dW_{k_\ell }(r) \bigg |^2 ds \bigg )^q\\\le & {} 2^q \mathbb {E}\bigg ( \frac{1}{\tau _{n_\ell }}\int _0^t \bigg | \int _{\kappa _{n_\ell }(s)}^s a_\ell (\bar{u}_\ell (r)) dr \bigg |^2 + \bigg |\int _{\kappa _{n_\ell }(s)}^s \varPi _{m_\ell } B \bar{u}_\ell (r) dW_{k_\ell }(r) \bigg |^2 ds \bigg )^q. \end{aligned}$$

Applying Hölder’s inequality yields

$$\begin{aligned} I_{1,\ell }(t)\le & {} 2^q \mathbb {E}\Bigg ( \frac{1}{\tau _{n_\ell }}\int _0^t\Bigg [ (s-\kappa _{n_\ell }(s)) \int _{\kappa _{n_\ell }(s)}^s |a_\ell (\bar{u}_\ell (r))|^2 dr \\&+\, \bigg |\int _{\kappa _{n_\ell }(s)}^s \varPi _{m_\ell } B \bar{u}_\ell (r) dW_{k_\ell }(r) \bigg |^2 \Bigg ]ds \Bigg )^q\\\le & {} c_q \mathbb {E}\bigg ( \frac{1}{\tau _{n_\ell }}\int _0^t \tau _{n_\ell }^2 |a_\ell (\bar{u}_\ell (s))|^2 ds \bigg )^q \\&+\, c_q\mathbb {E}\bigg ( \frac{1}{\tau _{n_\ell }}\int _0^t \bigg |\int _{\kappa _{n_\ell }(s)}^s \varPi _{m_\ell } B \bar{u}_\ell (r) dW_{k_\ell }(r) \bigg |^2 ds \bigg )^q. \end{aligned}$$

Using Assumption 2 and (2.6) one obtains

$$\begin{aligned} I_{1,\ell }(t)\le & {} c_q \mathbb {E}\bigg (\int _0^t \tau _{n_\ell } [ 2\mathfrak {c}(m)\Vert A_1 \bar{u}_\ell (s)\Vert _{V_1^*}^2 + 2n_\ell ] ds \bigg )^q \nonumber \\&+\, c_q\mathbb {E}\bigg ( \int _0^t \frac{1}{\tau _{n_\ell }} \bigg |\int _{\kappa _{n_\ell }(s)}^s \varPi _{m_\ell } B \bar{u}_\ell (r) dW_{k_\ell }(r) \bigg |^2 ds \bigg )^q := I_{2,\ell }(t) + I_{3,\ell }(t).\nonumber \\ \end{aligned}$$

(3.4)

The Growth assumption on $A_1$ implies that

$$\begin{aligned} I_{2,\ell }(t) \le c_{T,q}\bigg (1 + (\mathfrak {c}(m)\tau _{n_\ell })^q\mathbb {E}\bigg (\int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^q \bigg ). \end{aligned}$$

Using Hölder’s inequality leads to

$$\begin{aligned} I_{3,\ell }(t) \le c_{T,q} \mathbb {E}\int _0^t \frac{1}{\tau _{n_\ell }^q} \bigg |\int _{\kappa _{n_\ell }(s)}^s \varPi _{m_\ell } B \bar{u}_\ell (r) dW_{k_\ell }(r) \bigg |^{2q} ds. \end{aligned}$$

Due to Remark 1 and the Growth assumption on B one observes that

$$\begin{aligned} I_{3,\ell }(t)\le & {} c_{T,q} \int _0^t \frac{1}{\tau _{n_\ell }^q} \mathbb {E}\bigg |\int _{\kappa _{n_\ell }(s)}^s \Vert B \bar{u}_\ell (r)\Vert _{L_2(U,H)}^2 dr \bigg |^q ds \\\le & {} c_{T,q}\mathbb {E}\int _0^t \frac{1}{\tau _{n_\ell }^q} (s-\kappa _{n_\ell }(s))^q \Vert B \bar{u}_\ell (s)\Vert _{L_2(U,H)}^{2q} ds\\\le & {} c_{T,q}\bigg (1+\mathbb {E}\int _0^t |\bar{u}_\ell (s)|^{2q} ds \bigg ). \end{aligned}$$

This implies (3.2). Moreover Assumption 2 and (2.6) imply that

$$\begin{aligned} I_\ell (t):= & {} \mathbb {E}\bigg (\int _0^t |a_\ell (\bar{u}_\ell (s))||u_\ell (s)-\bar{u}_\ell (s)|ds\bigg )^q \\\le & {} 2^q \mathbb {E}\bigg (\int _0^t \tau _{n_\ell }|a_\ell (\bar{u}_\ell (s))|^2 + \frac{1}{\tau _{n_\ell }}|u_\ell (s)-\bar{u}_\ell (s)|^2 ds\bigg )^q\\\le & {} c_q \mathbb {E}\bigg (\int _0^t [\mathfrak {c}(m)\tau _{n_\ell }\Vert A_1 \bar{u}_\ell (s)\Vert _{V_1^*}^2 + \tau _{n_\ell } n_\ell ]ds \bigg )^q \\&+\, c_q \mathbb {E}\bigg (\frac{1}{\tau _{n_\ell }} \int _0^t |u_\ell (s)-\bar{u}_\ell (s)|^2 ds\bigg )^q. \end{aligned}$$

Applying the Growth assumption on $A_1$ yields

$$\begin{aligned} I_\ell (t)\le & {} c_{T,q}\bigg (1 + (\mathfrak {c}(m)\tau _{n_\ell })^q \mathbb {E}\bigg (\int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^q \bigg ) \nonumber \\&+\, c_q \mathbb {E}\bigg (\frac{1}{\tau _{n_\ell }} \int _0^t |u_\ell (s)-\bar{u}_\ell (s)|^2 ds\bigg )^q. \end{aligned}$$

(3.5)

Using (3.2) in (3.5) concludes the proof. $\square$

Theorem 3

(A priori estimate) Let the Coercivity and Growth conditions in Assumption 1 hold. Let Assumption 2 be satisfied. Let $q \ge 1$ be given and assume that $\mathbb {E}|u_\ell (0)|^{2q} < c$ and that $u_\ell (0)$ is $\mathcal {F}_0$-measurable. There is $\epsilon \in (0,\infty )$ such that for all $\ell \in \{\ell ' \in \mathbb {N}: \mathfrak {c}(m_{\ell '})\tau _{n_{\ell '}} < \epsilon \}$ we have,

for any $t\in [0,T]$,

$$\begin{aligned} \mathbb {E}\sup _{s\in [0,t]}|u_\ell (s)|^{2q} + \mu ^{q} \mathbb {E}\bigg (\int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^{q} \le c\left( 1+ \mathbb {E}|u_\ell (0)|^{2q}\right) . \end{aligned}$$

Proof

Applying the Coercivity assumption in (3.1), raising to power $q \ge 1$, taking the supremum over $s\le t$ and taking the expectation yields

$$\begin{aligned}&\mathbb {E}\sup _{s\le t} |u_\ell (s)|^{2q} + \mu ^{q} \mathbb {E}\bigg ( \int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^{q} \le c_{T,q} \bigg [1 + \mathbb {E}|u_\ell (0)|^{2q} \nonumber \\&\quad +\, \mathbb {E}\bigg ( \int _0^t |\bar{u}_\ell (s)|^2 ds \bigg )^{q} + \mathbb {E}\bigg (\int _0^t |a_\ell (\bar{u}_\ell (s))||u_\ell (s)-\bar{u}_\ell (s)|ds\bigg )^{q}\nonumber \\&\quad +\, \mathbb {E}\sup _{s \le t}\bigg |\int _0^s (u_\ell (s), B\bar{u}_\ell (s) dW_{k_\ell }(s))\bigg |^{q} \bigg ]. \end{aligned}$$

(3.6)

Using Lemma 1 in (3.6) results in

$$\begin{aligned}&\mathbb {E}\sup _{s\le t} |u_\ell (s)|^{2q} + \frac{\mu ^{q}}{2} \mathbb {E}\bigg ( \int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^{q}\nonumber \\&\quad \le c_{T,q} \bigg [1 + \mathbb {E}|u_\ell (0)|^{2q} + \mathbb {E}\bigg ( \int _0^t |\bar{u}_\ell (s)|^2 ds \bigg )^{q} \nonumber \\&\qquad +\, \mathbb {E}\int _0^t |\bar{u}_\ell (s)|^{2q} ds + \mathbb {E}\sup _{s \le t}\bigg (\int _0^s (u_\ell (r), B\bar{u}_\ell (r) dW_{k_\ell }(r))\bigg )^{q}\bigg ]. \end{aligned}$$

(3.7)

Using Burkholder–Davis–Gundy inequality one obtains

$$\begin{aligned} I_\ell:= & {} c_{T,q}\mathbb {E}\sup _{s \le t}\bigg |\int _0^s (u_\ell (r), B\bar{u}_\ell (r) dW_{k_\ell }(r))\bigg |^{q} \\\le & {} c_{T,q} \mathbb {E}\bigg |\int _0^t |u_\ell (s)|^2 \Vert B\bar{u}_\ell (s)\Vert _{L_2(U,H)}^2 ds\bigg |^{q/2}\\\le & {} c_{T,q} \mathbb {E}\Bigg [\sup _{s\le t}|u_\ell (s)|^{q}\bigg (\int _0^t \Vert B\bar{u}_\ell (s)\Vert _{L_2(U,H)}^2ds\bigg )^{q/2} \Bigg ]. \end{aligned}$$

Young’s inequality and the Growth assumption on B imply that

$$\begin{aligned} I_\ell\le & {} \frac{1}{2}\mathbb {E}\sup _{s\le t}|u_\ell (s)|^{2q} + c \mathbb {E}\bigg (\int _0^t \Vert B\bar{u}_\ell (s)\Vert _{L_2(U,H)}^2ds\bigg )^{q}\\\le & {} \frac{1}{2}\mathbb {E}\sup _{s\le t}|u_\ell (s)|^{2q} + c\bigg (1+\int _0^t \mathbb {E}\sup _{r\le s}|u_\ell (r)|^{2q} ds \bigg ). \end{aligned}$$

Applying this in (3.7) leads to

$$\begin{aligned}&\frac{1}{2}\mathbb {E}\sup _{s\le t} |u_\ell (s)|^{2q} + \frac{\mu ^{q}}{2} \mathbb {E}\bigg ( \int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^{q} \le c \bigg [1 + \mathbb {E}|u_\ell (0)|^{2q} \\&\quad + \int _0^t \mathbb {E}\sup _{r\le s}|u_\ell (r)|^{2q} ds\bigg ]. \end{aligned}$$

Application of Gronwall’s lemma yields

$$\begin{aligned} \mathbb {E}\sup _{s\in [0,t]}|u_\ell (s)|^{2q} + \mu ^{q} \mathbb {E}\bigg (\int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^{q} \le c \left( 1+\mathbb {E}|u_\ell (0)|^{2q}\right) . \end{aligned}$$

(3.8)

$\square$

Now we use Theorem 3 and Assumption 3 to obtain the remaining required estimates.

Corollary 1

(Remaining a priori estimates) Let the Growth and Coercivity conditions in Assumption 1 be satisfied. Let Assumptions 2 and 3 hold. Let $u_\ell (0)$ be bounded in $L^{q_0}(\varOmega ; H)$, uniformly with respect to $\ell$. There is $\epsilon \in (0,\infty )$ such that for all $\ell \in \{\ell ' \in \mathbb {N}: \mathfrak {c}({m_\ell '})\tau _{n_{\ell '}} < \epsilon \}$ we have

$$\begin{aligned} \mathbb {E}\int _0^T \Vert A_1 \bar{u}_\ell \Vert _{V_1^*}^2 ds \le c,\quad \mathbb {E}\int _0^T \Vert B \bar{u}_\ell \Vert _{L_2(U,H)}^2 ds \le c. \end{aligned}$$

(3.9)

Furthermore

$$\begin{aligned} \mathbb {E}\int _0^T |u_\ell (s) - \bar{u}_\ell (s)|^2 ds \le c \tau _{n_\ell }. \end{aligned}$$

(3.10)

Finally, for some $\eta > 0$,

$$\begin{aligned} \mathbb {E}\int _0^T \Vert \bar{u}_\ell \Vert _{V_2}^{p(1+\eta )} ds \le c \end{aligned}$$

(3.11)

and

$$\begin{aligned} \mathbb {E}\int _0^T \Vert A_2 \bar{u}_\ell \Vert _{V_2^*}^{p^*(1+\eta )} ds \le c,\quad \mathbb {E}\int _0^T \Vert A_{2,\ell } \bar{u}_\ell \Vert _{V_2^*}^{p^*(1+\eta )} ds \le c. \end{aligned}$$

(3.12)

Proof

Inequality (3.9) follows directly from the Growth assumptions on $A_1$ and B and from Theorem 3 with $q=1$. Using (3.2), together with Theorem 3 with $q=1$, yields (3.10). Since $u_\ell (0)$ is assumed to be bounded in $L^{q_0}(\varOmega ; H)$, uniformly in $\ell$, one can conclude, using Theorem 3, that

$$\begin{aligned} \mathbb {E}\sup _{s\in [0,t]}|u_\ell (s)|^{2\rho } \le c \,\,\text { and }\,\, \mathbb {E}\bigg (\int _0^t \Vert \bar{u}_\ell (s)\Vert _{V_1}^2 ds \bigg )^2 \le c. \end{aligned}$$

This, together with Assumption 3, yields (3.11). Finally, (3.11), the assumption on the growth of $A_2$ and (2.7) lead to (3.12). $\square$

4 Convergence

Having obtained the required a priori estimates we can use compactness arguments to extract weakly convergent subsequences of the approximation.

Lemma 2

Let the Growth and Coercivity conditions in Assumption 1 hold. Let Assumptions 2 and 3 be satisfied. Let $u_\ell (0) \rightarrow u_0$ in $L^{q_0}(\varOmega ; H)$. Let $\tfrac{\mathfrak {c}(m_\ell )}{n_\ell } \rightarrow 0$ as $\ell \rightarrow \infty$. Then there is a subsequence of the sequence $\ell$, which we denote $\ell '$, and $u \in \mathcal {L}^2(V_1) \cap \mathcal {L}^p(V_2)$ such that, as $\ell ' \rightarrow \infty$,

$$\begin{aligned} \bar{u}_{\ell '} \rightharpoonup u\,\, \text { in }\,\, \mathcal {L}^2(V_1)\,\, \text { and in }\,\, \mathcal {L}^p(V_2). \end{aligned}$$

Furthermore there are $a_1^\infty \in \mathcal {L}^2(V_1^*)$, $a_2^\infty \in \mathcal {L}^{p^*}(V_2^*)$ and $b^\infty \in \mathcal {L}^2(L_2(U,H))$ such that, as $\ell ' \rightarrow \infty$,

$$\begin{aligned} A_1\bar{u}_{\ell '} \rightharpoonup a_1^\infty \,\, \text { in }\,\, \mathcal {L}^2(V_1^*),\,\, A_{2,\ell '}\bar{u}_{\ell '} \rightharpoonup a_2^\infty \,\, \text { in }\,\, \mathcal {L}^{p^*}(V_2^*) \end{aligned}$$

and

$$\begin{aligned} B\bar{u}_{\ell '} \rightharpoonup b^\infty \,\, \text { in } \mathcal {L}^2({L_2(U,H)}). \end{aligned}$$

Finally, there is $\xi \in L^{q_0}(\varOmega ; H)$ such that $\bar{u}_{\ell '}(T) = u_{\ell '}(T) \rightharpoonup \xi$ in $L^{q_0}(\varOmega ; H)$.

Proof

The sequence $(\bar{u}_\ell )$ is bounded in $\mathcal {L}^2(V_1)$ due to Theorem 3 and in $\mathcal {L}^p(V_2)$ due to Corollary 1. The sequences $(A_1 \bar{u}_\ell )$, $(A_{2,\ell } \bar{u}_\ell )$ and $(B\bar{u}_\ell )$ are bounded in $\mathcal {L}^2(V_1^*)$, $\mathcal {L}^{p^*}(V_2^*)$ and in $\mathcal {L}^{2}(L_2(U,H))$ respectively, due to Corollary 1. Finally the sequence $(u_\ell (T))$ is bounded in $L^{q_0}(\varOmega ; H)$ due to Theorem 3.

Since it is assumed that $V_1$, $V_2$ are reflexive, it follows that $\mathcal {L}^2(V_1)$ and $\mathcal {L}^p(V_2)$ are reflexive. A bounded sequence in a reflexive Banach space must have a weakly convergent subsequence (see e.g. Brézis [2, Theorem 3.18]). Applying this to the sequences in question concludes the proof of the lemma. $\square$

Let $a^\infty := a_1^\infty + a_2^\infty$. Then $a^\infty \in \mathcal {L}^{p^*}(V^*)$. Due to Lemma 2 $a_\ell (\bar{u}_\ell ) \rightharpoonup a^\infty$ in $\mathcal {L}^{p^*}(V^*)$ as $\ell ' \rightarrow \infty$, provided that $\tfrac{\mathfrak {c}(m_\ell )}{n_\ell } \rightarrow 0$. The following lemma provides the equation satisfied by the weak limits of the approximations.

Lemma 3

Let the Growth and Coercivity conditions in Assumption 1 hold. Let Assumptions 2 and 3 be satisfied. Let $u_\ell (0) \rightarrow u_0$ in $L^{q_0}(\varOmega ; H)$. Let $\tfrac{\mathfrak {c}(m_\ell )}{n_\ell } \rightarrow 0$ as $\ell \rightarrow \infty$. Then there is an H-valued adapted continuous process $\tilde{u}$ on [0, T] such that $u=\tilde{u}$ $dt\times \mathbb {P}$-almost everywhere on $(0,T)\times \varOmega$. Furthermore, for almost every $(t,\omega ) \in (0,T) \times \varOmega$,

$$\begin{aligned} \tilde{u}(t) = u_0 + \int _0^t a^\infty (s) ds + \int _0^t b^\infty (s) dW(s) \end{aligned}$$

(4.1)

and almost surely

$$\begin{aligned} \tilde{u}(T) = u_0 + \int _0^T a^\infty (s) ds + \int _0^T b^\infty (s) dW(s). \end{aligned}$$

(4.2)

In the rest of this paper we will write u instead of $\tilde{u}$ for notational simplicity.

Proof

Fix $M\in \mathbb {N}$. Let $\varphi$ be a $V_M$-valued adapted stochastic process such that $|\varphi (t)| < M$ for all $t\in [0,T]$ and $\omega \in \varOmega$. For $g\in U$ let $\tilde{\varPi }_m g:= \sum _{j=1}^m (\chi _j,g)_U \chi _j$ and note that for any $v\in V$ one has $Bv\tilde{\varPi }_{m} \in L_2(U,H)$. From (2.2) one observes that

$$\begin{aligned} (u_{\ell '}(t),\varphi (t))= & {} (u_{\ell '}(0), \varphi (t)) + \bigg \langle \int _0^t a_{\ell '}(\bar{u}_{\ell '}(s)) ds, \varphi (t) \bigg \rangle \nonumber \\+ & {} \bigg (\int _0^t B \bar{u}_{\ell '}(s)\tilde{\varPi }_{m_{\ell '}}dW(s) ,\varphi (t)\bigg ). \end{aligned}$$

(4.3)

Let $G:\mathcal {L}^{p^*}(V^*) \rightarrow \mathcal {L}^{p^*}(V^*)$ be given by $(Gv)(t) := \int _0^t v(s)ds$. Moreover, let $I:\mathcal {L}^2(L_2(U,H)) \rightarrow \mathcal {L}^2(H)$ be given by $(Iv)(t) := \int _0^t v(s) dW(s)$. Integrating (4.3) from 0 to T and taking the expectation yields

$$\begin{aligned} \mathbb {E}\int _0^T (u_{\ell '}(t), \varphi (t)) dt= & {} \mathbb {E}\int _0^T (u_{\ell '}(0),\varphi (t))dt + \mathbb {E}\int _0^T \big \langle (G a_{\ell '}(\bar{u}_{\ell '}))(t), \varphi (t) \big \rangle dt\\&+\, \mathbb {E}\int _0^T \big ( (H \, B \bar{u}_{\ell '}\tilde{\varPi }_{m_{\ell '}} )(t) ,\varphi (t)\big ) dt. \end{aligned}$$

The operator G is linear and bounded and as such it is weakly-weakly continuous. This operator I is clearly linear. Furthermore, due to Itô’s isometry,

$$\begin{aligned} \Vert Iv \Vert _{\mathcal {L}^2(H)}^2= & {} \mathbb {E}\int _0^T |(Iv)(s)|^2 ds\\= & {} \int _0^T \mathbb {E}\bigg |\int _0^t v(s) dW(s)\bigg |^2 dt \\= & {} \mathbb {E}\int _0^T \int _0^t |v(s)|^2 ds dt \le T \Vert v\Vert _{\mathcal {L}^2(L_2(U,H))}^2. \end{aligned}$$

Thus the operator I is also bounded. It follows that I is also weakly-weakly continuous. Therefore, taking the limit as $\ell ' \rightarrow \infty$ and using Lemma 2, one obtains

$$\begin{aligned} \mathbb {E}\int _0^T (u(t), \varphi (t)) dt&= \mathbb {E}\int _0^T (u_0,\varphi (t))dt\\&\quad + \mathbb {E}\int _0^T \big \langle (G a^\infty )(t), \varphi (t) \big \rangle dt + \mathbb {E}\int _0^T \big ( (I b^\infty )(t) ,\varphi (t)\big ) dt. \end{aligned}$$

This holds for any $\varphi$ as specified at the beginning of the proof. By letting $M\rightarrow \infty$ and using the limited completeness of the Galerkin scheme it follows that this also holds for any $\varphi \in \mathcal {L}^{p}(V)$. Thus

$$\begin{aligned} u(t) = u_0 + \int _0^t a^\infty (s) ds + \int _0^t b^\infty (s) dW(s) \end{aligned}$$

(4.4)

holds for almost all $(t,\omega ) \in (0,T) \times \varOmega$.

Let $\varphi$ be a $V_M$-valued and $\mathcal {F}_T$-measurable random variable such that $\mathbb {E}\Vert \varphi \Vert _V^2 < \infty$. Setting $t=T$ in (4.3) and taking the expectation yields

$$\begin{aligned} \mathbb {E}(u_{\ell '}(T), \varphi )= & {} \mathbb {E}(u_{\ell '}(0),\varphi )\\&+\, \mathbb {E}\big \langle (G a_{\ell '}(\bar{u}_{\ell '}))(T), \varphi \big \rangle + \mathbb {E}\big ( (I \,B\bar{u}_{\ell '}\tilde{\varPi }_{m'} )(T) ,\varphi \big ). \end{aligned}$$

Let $\ell ' \rightarrow \infty$. The weak-weak continuity of the operators G and I, together with Lemma 2, implies that

$$\begin{aligned} \mathbb {E}(\xi , \varphi ) = \mathbb {E}(u_0,\varphi ) + \mathbb {E}\big \langle (G a^\infty )(T), \varphi \big \rangle + \mathbb {E}\big ( (I b^\infty )(T) ,\varphi \big ). \end{aligned}$$

(4.5)

Letting $M\rightarrow \infty$ and again using the limited completeness of the Galerkin scheme shows that the above equality holds for any $\mathcal {F}_T$-measurable $\varphi \in L^2(\varOmega ;V)$. If one now applies Itô’s formula to (4.4) then one obtains an adapted process $\tilde{u}$ with paths in C([0, T]; H) that is equal to u almost surely. Furthermore, for any $\varphi \in L^2(\varOmega ; V)$ and due to continuity of $\tilde{u}$,

$$\begin{aligned} \mathbb {E}(\xi - \tilde{u}(T),\varphi )= & {} \lim _{t\rightarrow T}\mathbb {E}(\xi - \tilde{u}(t),\varphi ) \\= & {} \lim _{t\rightarrow T} \mathbb {E}\bigg \langle \int _t^T a^\infty (s) ds + \int _t^T b^\infty (s)dW(s),\varphi \bigg \rangle = 0. \end{aligned}$$

Thus $\xi = \tilde{u}(T)$. This together with (4.5) implies (4.2). $\square$

All that remains to be done to prove Theorem 2 is to identify $a^\infty$ with Au and $b^\infty$ with Bu and to show strong convergence of $u_\ell (T)$ to u(T). To that end we would like to use monotonicity of A. In order to overcome the difficulty arising from the fact that the tamed operator $A_{2,\ell }$ does not preserve the monotonicity property of $A_2$ we need the following lemma.

Lemma 4

Let the Growth and Coercivity conditions in Assumption 1 hold. Let Assumptions 2 and 3 be satisfied. Let $u_\ell (0) \rightarrow u_0$ in $L^{q_0}(\varOmega ; H)$. Let $\tfrac{\mathfrak {c}(m_\ell )}{n_\ell } \rightarrow 0$ as $\ell \rightarrow \infty$. Then

$$\begin{aligned} \mathbb {E}\int _0^T \Vert A_2\bar{u}_\ell (s)-A_{2,\ell } \bar{u}_\ell (s)\Vert _{V_2^*}^{p^*} ds \rightarrow 0 \,\, \text { as }\,\, \ell \rightarrow \infty . \end{aligned}$$

Proof

Consider some $M>0$. Recall that $T_\ell$ is given by (2.5). Then

$$\begin{aligned} I_\ell:= & {} \mathbb {E}\int _0^T \Vert A_{2,\ell } \bar{u}_\ell (s) - A_2\bar{u}_\ell (s)\Vert _{V_2^*}^{p^*} ds \\= & {} \mathbb {E}\int _0^T \big (1-T_\ell (\bar{u}_\ell (s))\big )^{p^*}\Vert A_2 \bar{u}_\ell (s)\Vert _{V_2^*}^{p^*}\mathbbm {1}_{\{\Vert A_2 \bar{u}_\ell (s)\Vert _{V_2^*} \le M\}} ds\\&+\, \mathbb {E}\int _0^T \big (1-T_\ell (\bar{u}_\ell (s))\big )^{p^*}\Vert A_2 \bar{u}_\ell (s)\Vert _{V_2^*}^{p^*}\mathbbm {1}_{\{\Vert A_2 \bar{u}_\ell (s)\Vert _{V_2^*} > M\}} ds\\=: & {} I_{1,\ell ,M} + I_{2,\ell ,M}. \end{aligned}$$

It is observed that

$$\begin{aligned} I_{1,\ell ,M}\le & {} \mathbb {E}\int _0^T \frac{n_\ell ^{-1/2}|\varPi _{m_\ell } A_2 \bar{u}_\ell (s)|}{1+n_\ell ^{-1/2}|\varPi _{m_\ell } A_2 \bar{u}_\ell (s)|} \Vert A_2 \bar{u}_\ell (s)\Vert _{V_2^*}^{p^*}\mathbbm {1}_{\{\Vert A_2 \bar{u}_\ell (s)\Vert _{V_2^*} \le M\}} ds\\\le & {} \mathbb {E}\int _0^T \frac{\tau _{n_\ell }^{1/2}T^{-1/2} \mathfrak {c}(m)^{1/2}M}{1+n_\ell ^{-1/2}|\varPi _{m_\ell } A_2 \bar{u}_\ell (s)|} M^{p^*} ds \le (\mathfrak {c}(m)\tau _{n_\ell } )^{1/2} T^{1/2} M^{1+p^*}. \end{aligned}$$

Recall that due to Corollary 1 one knows that

$$\begin{aligned} \mathbb {E}\int _0^T \Vert A_2 \bar{u}_\ell (s)\Vert _{V_2^*}^{p^*(1+\eta )} ds < c \end{aligned}$$

with c independent of $\ell$. Thus the sequence $\big (\Vert A_2 \bar{u}_\ell \Vert _{V_2^*}^{p^*}\big )_{\ell \in \mathbb {N}}$ is uniformly integrable on $(0,T)\times \varOmega$ with respect to $dt \times P$. Hence for any $\epsilon > 0$ there exists M such that $I_{2,\ell ,M} < \epsilon /2$ for all $\ell$. Finally, since $\tfrac{\mathfrak {c}(m_\ell )}{n_\ell } \rightarrow 0$ as $\ell \rightarrow \infty$, one can choose $\ell$ large such that $I_{1,\ell ,M} < \epsilon /2$. $\square$

We now prove Theorem 1. This is needed to later show that the whole sequence of approximations converges rather than just a subsequence.

Proof of Theorem 1

Assume that $u_1$ and $u_2$ are two distinct solutions to (1.1) such that $u_1(0) = u_2(0) = u_0$. One would now like to apply Itô’s formula for the square of the norm from Pardoux [13, Chapitre 2,Theoreme 5.2]. To that end one immediately observes that $u_1 - u_2 \in \mathcal {L}^2(V_1) \cap \mathcal {L}^p(V_2)$ and that $u_1(0) - u_2(0) = 0 \in L^2(\varOmega ;H)$. Moreover

$$\begin{aligned} \Vert Au_1 - Au_2\Vert _{\mathcal {L}^2(V_1^*) + \mathcal {L}^{p^*}(V_2^*)} = \Vert A_1 u_1 - A_1 u_2\Vert _{\mathcal {L}^2(V_1^*)} + \Vert A_2 u_1 - A_2 u_2\Vert _{\mathcal {L}^{p^*}(V_2^*)}. \end{aligned}$$

Using the Growth assumption on $A_1$ one observes that

$$\begin{aligned}&\mathbb {E}\int _0^T \Vert A_1 u_1(s) - A_1 u_2(s)\Vert _{V_1^*}^2 ds \le \mathbb {E}\int _0^T 2\big [\Vert A_1 u_1(s)\Vert _{V_1^*}^2 + \Vert A_1 u_2(s)\Vert _{V_1^*}^2 \big ] ds\\&\quad \le K \mathbb {E}\int _0^T 2\big [(1+\Vert u_1(s)\Vert _{V_1}^2) + (1+\Vert u_2(s)\Vert _{V_1}^2) \big ] ds\\&\quad \le c(1 + \Vert u_1\Vert _{\mathcal {L}^2(V_1)}^2 + \Vert u_2\Vert _{\mathcal {L}^2(V_1)}^2) < \infty . \end{aligned}$$

Also, using the Growth assumption on $A_2$ one obtains

$$\begin{aligned} \mathbb {E}\int _0^T \Vert A_2 u_1(s) - A_2 u_2(s)\Vert _{V_2^*}^{p^*} ds \le c\left( 1 + \Vert u_1\Vert _{\mathcal {L}^p(V_2)}^p + \Vert u_2\Vert _{\mathcal {L}^p(V_2)}^p\right) < \infty . \end{aligned}$$

Thus $Au_1 - Au_2 \in \mathcal {L}^2(V_1^*)\cup \mathcal {L}^{p^*}(V_2^*)$. Finally, using the Growth assumption on B one deduces that $Bu_1 - Bu_2 \in \mathcal {L}^2(L_2(U,H))$. Hence the afromentioned Itô’s formula for the square of the norm can be applied, yielding

$$\begin{aligned} |u_1(t) - u_2(t)|^2= & {} -K \int _0^t e^{-Ks} |u_1(s) - u_2(s)|^2 ds\\&+ \int _0^t e^{-Ks}\big [ 2\langle Au_1(s) - Au_2(s),u_1(s)-u_2(s)\rangle \\&+\, \Vert Bu_1(s) - Bu_2(s)\Vert _{L_2(U,H)}^2 \big ] ds + M(t), \end{aligned}$$

where

$$\begin{aligned} M(t) := \int _0^t e^{-Ks}\big (u_1(s)-u_2(s),(Bu_1(s)-Bu_2(s))dW(s)\big ). \end{aligned}$$

One then observes, due to the monotonicity of $A:V\times \varOmega \rightarrow V^*$, that

$$\begin{aligned} |u_1(t)-u_2(t)|^2 \le M(t) \end{aligned}$$

and hence M is non-negative. It is also a real-valued continuous local martingale, and thus a supermartingale. Furthermore it starts from 0 and thus, almost surely, $M(t) = 0$ for all $t\in [0,T]$. One thus concludes that $u_1(t) = u_2(t)$ for all $t\in [0,T]$ almost surely. $\square$

Finally we can prove Theorem 2.

Proof of Theorem 2

Recall that $a_\ell (v) := \varPi _{m_\ell } [A_1 v + A_{2,\ell } v]$. Applying Itô’s formula to the scheme (2.2) and taking expectations yields

$$\begin{aligned} e^{-KT} \mathbb {E}|u_{\ell '}(T)|^2= & {} \mathbb {E}|u_{\ell '}(0)|^2 -K \mathbb {E}\int _0^T e^{-Ks} |u_{\ell '}(s)|^2 ds \\&+\, \mathbb {E}\int _0^T e^{-Ks}\Big ( 2 \langle a_{\ell '}(\bar{u}_{\ell '}(s)), \bar{u}_{\ell '}(s) \rangle + \Vert \varPi _{m_{\ell '}} B \bar{u}_{\ell '}(s) \Vert _{L_2(U,H)}^2 \Big ) ds\\&+\, \mathbb {E}\int _0^T 2e^{-Ks} \langle a_{\ell '}(\bar{u}_{\ell '}(s)), u_{\ell '}(s) - \bar{u}_{\ell '}(s) \rangle ds. \end{aligned}$$

Let

$$\begin{aligned} I_{1,\ell '} := \mathbb {E}\int _0^T \langle a_{\ell '}(\bar{u}_{\ell '}(s)), u_{\ell '}(s) - \bar{u}_{\ell '}(s) \rangle ds. \end{aligned}$$

Using Hölder’s inequality results in

$$\begin{aligned} I_{1,\ell '} \le \bigg (\mathbb {E}\int _0^T |a_\ell ( \bar{u}_{\ell '}(s))|^2 ds \bigg )^{1/2}\bigg (\mathbb {E}\int _0^T |u_{\ell '}(s) - \bar{u}_{\ell '}(s)|^2 ds \bigg )^{1/2}. \end{aligned}$$

Using Assumption 2 and Corollary 1 yields

$$\begin{aligned} I_{1,\ell '} \le c (\mathfrak {c}(m_{\ell '})\tau _{n_{\ell '}})^{1/2}. \end{aligned}$$

Thus,

$$\begin{aligned} e^{-Ks}\mathbb {E}|u_{\ell '}(T)|^2\le & {} \mathbb {E}|u_{\ell '}(0)|^2 -K \mathbb {E}\int _0^T e^{-Ks}|u_{\ell '}(s)|^2 ds\\&+ \mathbb {E}\int _0^T e^{-Ks}\big ( 2 \langle a_{\ell '}(\bar{u}_{\ell '}(s)), \bar{u}_{\ell '}(s) \rangle + \Vert B \bar{u}_{\ell '}(s)\Vert _{L_2(U,H)}^2 \big ) ds\\&+ \,c(\mathfrak {c}(m_{\ell '})\tau _{n_{\ell '}})^{1/2} \end{aligned}$$

and one may proceed with a monotonicity argument. Let $w\in \mathcal {L}^p(V)$. Then

$$\begin{aligned}&e^{-KT}\mathbb {E}|u_{\ell '}(T)|^2 \le \mathbb {E}|u_{\ell '}(0)|^2-K \mathbb {E}\int _0^T e^{-Ks}|u_{\ell '}(s)|^2 ds\\&\quad +\, \mathbb {E}\int _0^T e^{-Ks}\big [ 2 \langle a_{\ell '}(\bar{u}_{\ell '}(s)) - a_{\ell '}(w(s)), \bar{u}_{\ell '}(s) - w(s)\rangle \\&\quad +\, 2 \langle a_{\ell '}(w(s)), \bar{u}_{\ell '}(s)) - w(s)\rangle +\, 2\langle a_{\ell '}(\bar{u}_{\ell '}(s)), w(s) \rangle \\&\quad +\, 2(Bw(s),B\bar{u}_{\ell '}(s))_{L_2(U,H)\times L_2(U,H)} - \Vert Bw(s)\Vert _{L_2(U,H)}^2\\&\quad +\, \Vert B\bar{u}_{\ell '}(s)) - Bw(s)\Vert _{L_2(U,H)}^2 \big ] ds + c(\mathfrak {c}(m_{\ell '})\tau _{n_{\ell '}})^{1/2}. \end{aligned}$$

Using the Monotonicity assumption on A one obtains

$$\begin{aligned} e^{-KT}\mathbb {E}|u_{\ell '}(T)|^2\le & {} \mathbb {E}|u_{\ell '}(0)|^2\nonumber \\&+\, K\mathbb {E}\int _0^T e^{-Ks} \big (|\bar{u}_{\ell '}(s)|^2 -|u_{\ell '}(s)|^2 -2(\bar{u}_{\ell '}(s),w(s))\nonumber \\&\qquad + |w(s)|^2 \big ) ds \nonumber \\&+\, \mathbb {E}\int _0^T e^{-Ks} \langle A_{2,\ell '}\bar{u}_{\ell '}(s) - A_2 \bar{u}_{\ell '}(s), \bar{u}_{\ell '}(s) - w(s)\rangle ds\nonumber \\&+\, \mathbb {E}\int _0^T e^{-Ks} \langle A_2 w(s) - A_{2,\ell '}w(s), \bar{u}_{\ell '}(s) - w(s)\rangle ds\nonumber \\&+\, \mathbb {E}\int _0^T e^{-Ks} \big [ 2 \langle a_{\ell '}(w(s)), \bar{u}_{\ell '}(s)) - w(s)\rangle \nonumber \\&+\, 2\langle a_{\ell '}(\bar{u}_{\ell '}(s)), w(s) \rangle \nonumber \\&+\, 2(Bw(s),B\bar{u}_\ell (s))_{L_2(U,H)\times L_2(U,H)} - \Vert Bw(s)\Vert _{L_2(U,H)}^2 \big ] ds\nonumber \\&+\, c(\mathfrak {c}(m_{\ell '})\tau _{n_{\ell '}})^{1/2}. \end{aligned}$$

(4.6)

Taking limit inferior as $\ell ' \rightarrow \infty$, using the weak lower-semi-continuity of the norm, Lemmas 2, 4 and Corollary 1, one observes that

$$\begin{aligned} e^{-KT}\mathbb {E}|u(T)|^2\le & {} \mathbb {E}|u_0|^2 + K\mathbb {E}\int _0^T e^{-Ks} \big [ -2(u(s),w(s)) + |w(s)|^2 \big ] ds \nonumber \\&+\, \mathbb {E}\int _0^T e^{-Ks}\big [ 2 \langle A w(s), u(s) - w(s) \rangle +\, 2\langle a^\infty (s), w(s) \rangle \nonumber \\&+\, 2(Bw(s), b^\infty (s))_{L_2(U,H)\times L_2(U,H)} - \Vert Bw(s)\Vert _{L_2(U,H)}^2 \big ] ds.\nonumber \\ \end{aligned}$$

(4.7)

Applying Itô’s formula to (4.1) and taking expectations yields

$$\begin{aligned} e^{-KT}\mathbb {E}|u(T)|^2= & {} \mathbb {E}|u(0)|^2 - K\mathbb {E}\int _0^T e^{-Ks} |u(s)|^2 ds\nonumber \\&+\, \mathbb {E}\int _0^T e^{-Ks} \big [2\langle a^\infty (s),u(s)\rangle + \Vert b^\infty (s)\Vert _{L_2(U,H)}^2 \big ]ds \end{aligned}$$

(4.8)

Subtracting this from (4.7) one arrives at

$$\begin{aligned} 0\le & {} \mathbb {E}\int _0^T e^{-Ks}\big [ K|u(s)-w(s)|^2 + 2 \langle A w(s), u(s) - w(s) \rangle \nonumber \\&+\, 2 \langle a^\infty (s), w(s) - u(s)\rangle - \Vert Bu(s) - b^\infty (s)\Vert _{L_2(U,H)}^2 \big ]ds. \end{aligned}$$

(4.9)

Note that so far w was arbitrary. It will now be used to identify the nonlinear terms. First, one takes $w=u$ and observes that,

$$\begin{aligned} 0 \le - \mathbb {E}\int _0^T e^{-Ks} \Vert Bu(s) - b^\infty (s)\Vert _{L_2(U,H)}^2 ds \le 0 \end{aligned}$$

which implies $b^\infty = Bu$. Next, one sets $w=u+\epsilon z$ with $\epsilon > 0$ and $z\in \mathcal {L}^p(V)$ in (4.9). Dividing by $\epsilon > 0$ leads to

$$\begin{aligned} 0 \le \mathbb {E}\int _0^T e^{-Ks} \big [ K\epsilon |z|^2+ 2 \langle A (u(s)+\epsilon z(s)), -z(s) \rangle + 2 \langle a^\infty (s), z(s)\rangle \big ]ds. \end{aligned}$$

Using hemicontinuity of A while letting $\epsilon \rightarrow 0$ results in

$$\begin{aligned} \mathbb {E}\int _0^T \langle A u(s), z(s) \rangle ds \le \mathbb {E}\int _0^T \langle a^\infty (s), z(s)\rangle ds. \end{aligned}$$

This holds for an arbitrary $z\in \mathcal {L}^p(V)$ and hence, in particular, for $-z$. Thus one obtains that $a^\infty = Au$.

Due to Theorem 1, the solution u to (1.1) is unique. Thus the whole sequences of approximations converges rather than just the subsequence denoted by $\ell '$.

Finally, in order to show that $u_\ell (T) \rightarrow u(T)$ in $L^2(\varOmega ;H)$, one uses (4.6) and (4.7) with $w=u$ together with $a^\infty = Au$ and $b^\infty = Bu$. Consequently, the weak-lower-semi-continuity of the norm and Lemma 2 lead to

$$\begin{aligned} e^{-KT}\mathbb {E}|u(T)|^2\le & {} \liminf _{\ell \rightarrow \infty } e^{-KT} \mathbb {E}|u_\ell (T)|^2 \le \mathbb {E}|u_0|^2 -K \mathbb {E}\int _0^T e^{-Ks} |u(s)|^2 ds \\&+\,\mathbb {E}\int _0^T e^{-Ks}\big [2\langle Au(s),u(s) \rangle + \Vert Bu(s)\Vert _{L_2(U,H)}^2 \big ]ds. \end{aligned}$$

Thus, due to (4.8),

$$\begin{aligned} 0 \le \liminf _{\ell \rightarrow \infty } \mathbb {E}|u_\ell (T)|^2 - \mathbb {E}|u(T)|^2 \le 0. \end{aligned}$$

From Lemmas 2 and 3, one already knows that $u_\ell (T) \rightharpoonup u(T)$ in $L^2(\varOmega ;H)$. This is a uniformly convex space (as it is a Hilbert space). Thus one concludes that $u_\ell (T) \rightarrow u(T)$ in $L^2(\varOmega ;H)$. For this see, e.g., Brézis [2, Proposition 3.32]. $\square$

5 Examples

In this section we give examples of three equations which fit into our framework. In all three examples the interpolation inequality is a consequence of the Gagliardo–Nirenberg inequality (see, for example, [15, Theorem 1.24]). The first example is the equation:

$$\begin{aligned} du = \left[ \nabla a(\nabla u) - |u|^{p-2} u\right] dt + u dW\,\, \text { on } \,\, \mathscr {D}\times (0,T) \end{aligned}$$

with $u=0$ on the boundary of the domain $\mathscr {D}$ and $u(\cdot ,0) = u_0$ given. Here $a:\mathbb {R}^d \rightarrow \mathbb {R}^d$ can be nonlinear but it is assumed to be continuous, monotone and growing at most linearly. If we take $a_i(z) = z_i$ then $\nabla a(\nabla u) = \varDelta u$ and this equation is the stochastic Ginzburg–Landau equation. An example of a nonlinear function is $a_i(z) = \tfrac{2+\exp (-z_i)}{1+\exp (-z_i)}$. Moreover $\mathscr {D}$ is a bounded Lipschitz domain in $\mathbb {R}^d$, $d=1,2,3$ and $p\in [2,6)$ if $d=1$, $p\in [2,4)$ if $d=2$ and $p\in [2,10/3)$ if $d=3$. In our framework $H=L^2(\mathscr {D})$, $V_1 = H^1_0(\mathscr {D})$ and $V_2 = L^p(\mathscr {D})$ (using the standard notation for Lebesgue and Sobolev spaces).

The second is the stochastic Swift–Hohenberg equation:

$$\begin{aligned} du = \Big [\left( \gamma ^2 - (1+\varDelta )^2\right) u - |u|^{p-2}u\Big ]dt + dW\,\, \text { on } \,\, \mathscr {D}\times (0,T) \end{aligned}$$

with appropriate boundary and initial conditions. The domain $\mathscr {D}$ is assumed to be a bounded Lipschitz domain in $\mathbb {R}^2$. With Dirichlet boundary conditions we would take $V_1 = H^2_0(\mathscr {D})$ and $V_2 = L^p(\mathscr {D})$ with $p\in [2,6)$.

The third example is the spatially extended stochastic FitzHugh–Nagumo system for signal propagation in nerve cells (originally stated by FitzHugh [3] as a system of ordinary differential equations, see Bonaccorsi and Mastrogiacomo [1] for mathematical analysis of the spatially extended stochastic version):

$$\begin{aligned} \begin{array}{ll} du &{} = (\varDelta u + u - u^3 - v)dt + dW\\ dv &{} = c_1(u - c_2 v + c_3)dt \end{array} \,\, \text { on } \,\, (0,1)\times (0,T), \end{aligned}$$

together with appropriate initial data for u and v as well as homogeneous Neumann boundary conditions for u only. In this situation $V_1 = H^1((0,1))\times L^2((0,1))$ while $V_2 = L^4((0,1))\times L^2((0,1))$.

We now provide estimates on the constant $\mathfrak {c}(m)$ in the particular case when $\mathscr {D} = (0,\pi )^2 \subset \mathbb {R}^2$ and we use a spectral Galerkin method to construct the spaces $V_m$. To that end define

$$\begin{aligned} \varphi _{n_1 n_2}(x_1,x_2) := \frac{2}{\pi }\sin (n_1 x_1)\sin (n_2 x_2). \end{aligned}$$

Let $V_m = \text {span}\{\varphi _{n_1 n_2} : n_1 = 1,\ldots , m, n_2 = 1,\ldots ,m\}$. Then

$$\begin{aligned} \varPi _m f := \sum _{n_1=1, n_2=1}^m \langle f, \varphi _{n_1 n_2} \rangle \varphi _{n_1 n_2} \end{aligned}$$

satisfies Assumption 2. Moreover we can calculate

$$\begin{aligned} \mathfrak {c}(m)= & {} \sum _{n_1=1, n_2=1}^m \left( \Vert \varphi _{n_1 n_2}\Vert _{L^2(\mathscr {D})}^2 + \Vert \nabla \varphi _{n_1 n_2}\Vert _{L^2(\mathscr {D};\mathbb {R}^2)}^2 + \Vert \varphi _{n_1 n_2}\Vert _{L^p(\mathscr {D})}^{2/p} \right) \\= & {} m^2 \left( 1 + 2 + c_p\right) , \end{aligned}$$

where $c_p$ depends only on p. Hence, in order to apply Theorem 2, we need a sequence $(m_\ell , n_\ell , k_\ell )$ such that $\tfrac{m_\ell ^2}{n_\ell } \rightarrow 0$ as $\ell \rightarrow \infty$. This means that we need to choose $n_\ell = \lfloor m_\ell ^{2+\delta } \rfloor$ for some $\delta > 0$.

We also note that if $\mathscr {D} = (0,\pi )^d$ then an analogous construction of $V_m$ would lead to the conclusion that we need $n_\ell = \lfloor m_l^{d+\delta }\rfloor$ for some $\delta > 0$. Crucially we see that the space-time coupling requirement is no more onerous than in the case of equations with operators growing at most linearly.