Abstract
We consider a model initial- and Dirichlet boundary- value problem for a linearized Cahn–Hilliard–Cook equation, in one space dimension, forced by the space derivative of a space–time white noise. First, we introduce a canvas problem, the solution to which is a regular approximation of the mild solution to the problem and depends on a finite number of random variables. Then, fully discrete approximations of the solution to the canvas problem are constructed using, for discretization in space, a Galerkin finite element method based on \(H^2\) piecewise polynomials, and, for time-stepping, an implicit/explicit method. Finally, we derive a strong a priori estimate of the error approximating the mild solution to the problem by the canvas problem solution, and of the numerical approximation error of the solution to the canvas problem.
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1 Introduction
Let \(T>0\), \(D:=(0,1)\), and \((\Omega ,{\mathcal {F}},P)\) be a complete probability space. Then, we consider the model initial- and Dirichlet boundary- value problem for a linearized Cahn–Hilliard–Cook equation formulated in Kossioris and Zouraris (2013), which is as follows: find a stochastic function \(u:[0,T]\times {\overline{D}}\rightarrow {\mathbb {R}}\), such that
a.s. in \(\Omega \), where \({\dot{W}}\) denotes a space-time white noise on \([0,T]\times D\) (see, e.g., Walsh 1986; Kallianpur and Xiong 1995) and \(\mu \) is a real constant. We recall that the mild solution to the problem above (cf. Debussche and Zambotti 2007) is given by
where
\(\lambda _k:=k\,\pi \) for \(k\in {\mathbb {N}}\), \(\varepsilon _k(z):=\sqrt{2}\,\sin (\lambda _k\,z)\) and \(\varphi _k(z):=\sqrt{2}\,\cos (\lambda _k\,z)\) for \(z\in {\overline{D}}\) and \(k\in {\mathbb {N}}\), and \(\mathsf{G}_t(x,y)\) is the space-time Green kernel of the solution to the deterministic parabolic problem: find \(w:[0,T]\times {\overline{D}}\rightarrow {\mathbb {R}}\), such that
In the paper at hand, our goal is to propose and analyze a numerical method for the approximation of u that has less stability requirements and lower complexity than the method proposed in Kossioris and Zouraris (2013).
1.1 A canvas problem
A canvas problem is an initial- and boundary- value problem the solution to which: i) depends on a finite number of random variables and ii) is a regular approximation of the mild solution u to (1.1). Then, we can derive computable approximations of u by constructing numerical approximations of the canvas problem solution via the application of a discretization technique for stochastic partial differential equations with random coefficients. The formulation of the canvas problem depends on the way which we replace the infinite stochastic dimensionality of the problem (1.1) by a finite one.
In our case, the canvas problem is formulated as follows (cf. Allen et al. 1998; Kossioris and Zouraris 2010, 2013): Let \(\mathsf{M},\mathsf{N}\in {\mathbb {N}}\), \(\Delta {t}:=\frac{T}{\mathsf{N}}\), and \(t_n:=n\,\Delta {t}\) for \(n=0,\dots ,\mathsf{N}\), \(\mathsf{T}_n:=(t_{n-1},t_n)\) for \(n=1,\dots ,\mathsf{N}\), and \(\mathsf{u}:[0,T]\times {\overline{D}}\rightarrow {\mathbb {R}}\), such that
where
and \(B^i(t):=\int _0^t\int _{\scriptscriptstyle D}\varphi _i(x)\;\mathrm{d}W(s,x)\) for \(t\ge 0\) and \(i\in {\mathbb {N}}\). According to Walsh (1986), \((B^i)_{i=1}^{\infty }\) is a family of independent Brownian motions, and thus, the random variables \(\left( \left( R^n_i\right) _{n=1}^{\scriptscriptstyle \mathsf{N}}\right) _{i=1}^{\scriptscriptstyle \mathsf M}\) are independent and satisfy
Thus, the solution \(\mathsf{u}\) to (1.5) depends on \(\mathsf{N}\mathsf{M}\) random variables and the well-known theory for parabolic problems (see, e.g, Lions and Magenes 1972) yields its regularity along with the following representation formula:
Remark 1.1
In Kossioris and Zouraris (2013), the definition of \({\mathcal {W}}\) is based on a uniform partition of [0, T] in N subintervals and on a uniform partition of D in J subintervals. At every time-slab, \({\mathcal {W}}\) has a constant value with respect to the time variable, but, with respect to the space variable, is defined as the \(L^2(D)\)-projection of a random, piecewise constant function onto the space of linear splines, the computation of which leads to the numerical solution of a \((J+1)\times (J+1)\) tridiagonal linear system of algebraic equations. Finally, \({\mathcal {W}}\) depends on \(N (J+1)\) random variables and its construction has \(O(N\,(J+1))\) complexity, which must to be added to the complexity of the numerical method used for the approximation of \(\mathsf{u}\). On the contrary, the stochastic load \({\mathcal {W}}\) of the canvas problem (1.5) which we propose here is given explicitly by the formula (1.6), and thus, no extra computational cost is required for its formation.
1.2 An IMEX finite element method
Let \(M\in {\mathbb {N}}\), \(\Delta \tau :=\frac{T}{M}\), and \(\tau _m:=m\,\Delta \tau \) for \(m=0,\dots ,M\), and \(\Delta _m:=(\tau _{m-1},\tau _m)\) for \(m=1,\dots ,M\). In addition, for \(r=2\) or 3, let \(\mathsf{M}_h^r\subset H^2(D)\cap H_0^1(D)\) be a finite element space consisting of functions which are piecewise polynomials of degree at most r over a partition of D in intervals with maximum mesh length h.
The fully discrete method which we propose for the numerical approximation of \(\mathsf{u}\) uses an implicit/explicit (IMEX) time-discretization treatment of the space differential operator along with a finite element variational formulation for space discretization. Its algorithm is as follows: first, sets
and then, for \(m=1,\dots ,M\), finds \(\mathsf{U}_h^m\in \mathsf{M}_h^r\), such that
for all \(\chi \in \mathsf{M}_h^r\), where \((\cdot ,\cdot )_{\scriptscriptstyle 0,D}\) is the usual \(L^2(D)\)-inner product.
Remark 1.2
It is easily seen that the numerical method above is unconditionally stable, while the Backward Euler finite element method is stable under the time-step restriction: \(\Delta \tau \,\mu ^2\le 4\) (see Kossioris and Zouraris 2013).
1.3 An overview of the paper
In Sect. 2, we introduce notation and we recall several results that are often used in the rest of the paper. In Sect. 3, we focus on the estimation of the error which we made by approximating the solution u to (1.1) by the solution \(\mathsf{u}\) to (1.5), arriving at the bound
(see Theorem 3.1). Section 4 is dedicated to the definition and the convergence analysis of modified IMEX time-discrete and fully discrete approximations of the solution w to the deterministic problem (1.4). The results obtained are used later in Sect. 5, where we analyze the numerical method for the approximation of \(\mathsf{u}\), given in Sect. 1.2. Its convergence is established by proving the following strong error estimate:
for all \(\epsilon _1\in \left( 0,\frac{1}{8}\right] \) and \(\epsilon _2\in \left( 0,\frac{r}{6}\right] \) (see Theorem 5.3). We obtain the latter error bound, by applying a discrete Duhamel principle technique to estimate separately the time-discretization error and the space-discretization error, which are defined using as an intermediate the corresponding IMEX time-discrete approximations of \(\mathsf{u}\), specified by (5.1) and (5.2) (cf., e.g., Kossioris and Zouraris 2010, 2013; Yan 2005).
Since we have no assumptions on the sign, or, the size of \(\mu \), the elliptic operator in (1.5) is, in general, not invertible. This is the reason that the Backward Euler/finite element method is stable and convergent after adopting a restriction on the time-step size (see Kossioris and Zouraris 2013, Remark 1.2). On the contrary, the IMEX/finite element method which we propose here is unconditionally stable and convergent, because the principal part of the elliptic operator is treated implicitly and its lower order part explicitly. Another characteristic in our method is the choice to build up the canvas problem using spectral functions, which allow us to avoid the numerical solution of an extra linear system of algebraic equation at every time-step that is required in the approach of Kossioris and Zouraris (2013) (see Remark 1.1).
The error analysis of the IMEX finite element method is more technical than that in Kossioris and Zouraris (2013) for the Backward Euler finite element method. The main difference is due to the fact that the representation of the time-discrete and fully discrete approximations of \(\mathsf{u}\) is related to a modified version of the IMEX time-stepping method for the approximation of the solution to the deterministic problem (1.4), the error analysis of which is necessary in obtaining the desired error estimate and is of independent interest (see Sect. 4).
2 Preliminaries
We denote by \(L^2(D)\) the space of the Lebesgue measurable functions which are square integrable on D with respect to the Lebesgue measure \(\mathrm{d}x\). The space \(L^2(D)\) is provided with the standard norm \(\Vert g\Vert _{\scriptscriptstyle 0,D}:= \left( \int _{\scriptscriptstyle D}|g(x)|^2\,\mathrm{d}x\right) ^{\frac{1}{2}}\) for \(g\in L^2(D)\), which is derived by the usual inner product \((g_1,g_2)_{\scriptscriptstyle 0,D}:=\int _{\scriptscriptstyle D}g_1(x)\,g_2(x)\,\mathrm{d}x\) for \(g_1\), \(g_2\in L^2(D)\). In addition, we employ the symbol \({\mathbb {N}}_0\) for the set of all nonnegative integers.
For \(s\in {\mathbb {N}}_0\), we denote by \(H^s(D)\) the Sobolev space of functions having generalized derivatives up to order s in \(L^2(D)\), and by \(\Vert \cdot \Vert _{\scriptscriptstyle s,D}\) its usual norm, i.e., \(\Vert g\Vert _{\scriptscriptstyle s,D}:=\left( \sum _{\ell =0}^s \Vert \partial _x^{\ell }g\Vert _{\scriptscriptstyle 0,D}^2\right) ^{\scriptscriptstyle 1/2}\) for \(g\in H^s(D)\). In addition, by \(H_0^1(D)\), we denote the subspace of \(H^1(D)\) consisting of functions which vanish at the endpoints of D in the sense of trace.
The sequence of pairs \(\left\{ \left( \lambda _i^2,\varepsilon _i\right) \right\} _{i=1}^{\infty }\) is a solution to the eigenvalue/eigenfunction problem: find nonzero \(\varphi \in H^2(D)\cap H_0^1(D)\) and \(\lambda \in {\mathbb {R}}\), such that \(-\varphi ''=\lambda \,\varphi \) in D. Since \((\varepsilon _i)_{i=1}^{\infty }\) is a complete \((\cdot ,\cdot )_{\scriptscriptstyle 0,D}\)-orthonormal system in \(L^2(D)\), for \(s\in {\mathbb {R}}\), we define by
a subspace of \(L^2(D)\) provided with the natural norm \(\Vert v\Vert _{\scriptscriptstyle {\mathcal {V}}^s}:=\big (\,\sum _{i=1}^{\infty } \lambda _{i}^{2s}\,(v,\varepsilon _i)^2_{\scriptscriptstyle 0,D}\,\big )^{\scriptscriptstyle 1/2}\) for \(v\in {\mathcal {V}}^s(D)\). For \(s\ge 0\), the space \(({\mathcal {V}}^s(D),\Vert \cdot \Vert _{\scriptscriptstyle {\mathcal {V}}^s})\) is a complete subspace of \(L^2(D)\) and we define \(({\dot{\mathbf{H}}}^{{s}}(D),\Vert \cdot \Vert _{\scriptscriptstyle {\dot{\mathbf{H}}}^s}) :=({\mathcal {V}}^s(D),\Vert \cdot \Vert _{\scriptscriptstyle {\mathcal {V}}^s})\). For \(s<0\), the space \(({\dot{\mathbf{H}}}^s(D),\Vert \cdot \Vert _{\scriptscriptstyle {\dot{\mathbf{H}}}^s})\) is defined as the completion of \(({\mathcal {V}}^s(D),\Vert \cdot \Vert _{\scriptscriptstyle {\mathcal {V}}^s})\), or, equivalently, as the dual of \(({\dot{\mathbf{H}}}^{-s}(D),\Vert \cdot \Vert _{\scriptscriptstyle {\dot{\mathbf{H}}}^{-s}})\).
Let \(m\in {\mathbb {N}}_0\). It is well known (see Thomée 1997) that
and that there exist constants \(C_{m,{\scriptscriptstyle A}}\) and \(C_{m,{\scriptscriptstyle B}}\), such that
In addition, we define on \(L^2(D)\) the negative norm \(\Vert \cdot \Vert _{\scriptscriptstyle -m, D}\) by
for which, using (2.1), follows that there exists a constant \(C_{-m}>0\), such that:
Let \({\mathbb {L}}_2=(L^2(D),(\cdot ,\cdot )_{\scriptscriptstyle 0,D})\) and \({\mathcal {L}}({\mathbb {L}}_2)\) be the space of linear, bounded operators from \({\mathbb {L}}_2\) to \({\mathbb {L}}_2\). An operator \(\Gamma \in {\mathcal {L}}({\mathbb {L}}_2)\) is Hilbert–Schmidt, when \(\Vert \Gamma \Vert _{\scriptscriptstyle \mathrm HS}:=\left( \sum _{i=1}^{\infty } \Vert \Gamma \varepsilon _i\Vert ^2_{\scriptscriptstyle 0,D}\right) ^{\frac{1}{2}}<+\infty \), where \(\Vert \Gamma \Vert _{\scriptscriptstyle \mathrm HS}\) is the so-called Hilbert–Schmidt norm of \(\Gamma \). We note that the quantity \(\Vert \Gamma \Vert _{\scriptscriptstyle \mathrm HS}\) does not change when we replace \((\varepsilon _i)_{i=1}^{\infty }\) by another complete orthonormal system of \({\mathbb {L}}_2\). It is well known (see, e.g., Dunford and Schwartz 1988; Lord et al. 2014) that an operator \(\Gamma \in {\mathcal {L}}({\mathbb {L}}_2)\) is Hilbert–Schmidt iff there exists a measurable function \(\gamma :D\times D\rightarrow {\mathbb {R}}\), such that \(\Gamma [v](\cdot )=\int _{\scriptscriptstyle D}\gamma (\cdot ,y)\,v(y)\,dy\) for \(v\in L^2(D)\), and then, it holds that
Let \({\mathcal {L}}_{\scriptscriptstyle \mathrm HS}({\mathbb {L}}_2)\) be the set of Hilbert–Schmidt operators of \({\mathcal {L}}({\mathbb {L}}^2)\) and \(\Phi :[0,T]\rightarrow {\mathcal {L}}_{\scriptscriptstyle \mathrm HS}({\mathbb {L}}_2)\). In addition, for a random variable X, let \({\mathbb {E}}[X]\) be its expected value, i.e., \({\mathbb {E}}[X]:=\int _{\scriptscriptstyle \Omega }X\,\mathrm{d}P\). Then, the Itô isometry property for stochastic integrals reads
For later use, we recall that if \(({\mathcal {H}},(\cdot ,\cdot )_{\scriptscriptstyle {\mathcal {H}}})\) is a real inner product space with induced norm \(|\cdot |_{\scriptscriptstyle {\mathcal {H}}}\), then
Finally, for any nonempty set A, we denote by \({\mathcal {X}}_{\scriptscriptstyle A}\) the indicator function of A.
2.1 A projection operator
Let \({\mathcal {O}}:=(0,T)\times D\), \({\mathfrak S}_{\scriptscriptstyle \mathsf{M}}:={\mathrm{span}}(\varphi _i)_{i=1}^{\scriptscriptstyle \mathsf{M}}\), \({\mathfrak S}_{\scriptscriptstyle \mathsf{N}}:=\mathrm{span}({\mathcal {X}}_{\scriptscriptstyle T_n})_{n=1}^{\scriptscriptstyle \mathsf{N}}\) and \(\mathsf{\Pi }:L^2({\mathcal {O}})\rightarrow {\mathfrak S}_{\scriptscriptstyle \mathsf{N}}\otimes {\mathfrak S}_{\scriptscriptstyle \mathsf{M}}\), the usual \(L^2({\mathcal {O}})\)-projection operator which is given by the formula:
Then, the following representation of the stochastic integral of \(\mathsf{\Pi }\) holds [cf. Lemma 2.1 in Kossioris and Zouraris (2010)].
Lemma 2.1
For \(g\in L^2({\mathcal {O}})\), it holds that
Proof
Using (2.6) and (1.7), we have
which along (1.6) yields (2.7). \(\square \)
2.2 Linear elliptic and parabolic operators
Let \(T_{\scriptscriptstyle E}:L^2(D)\rightarrow {\dot{\mathbf{H}}}^2(D)\) be the solution operator of the Dirichlet two-point boundary-value problem: for given \(f\in L^2(D)\) find \(v_{\scriptscriptstyle E}\in {\dot{\mathbf{H}}}^2(D)\), such that \(v_{\scriptscriptstyle E}''=f\) in D, i.e., \(T_{\scriptscriptstyle E}f:=v_{\scriptscriptstyle E}\). It is well known that
and, for \(m\in {\mathbb {N}}_0\), there exists a constant \(C_{\scriptscriptstyle E}^m>0\), such that
Let, also, \(T_{\scriptscriptstyle B}:L^2(D)\rightarrow {\dot{\mathbf{H}}}^4(D)\) be the solution operator of the following Dirichlet biharmonic two-point boundary-value problem: for given \(f\in L^2(D)\) find \(v_{\scriptscriptstyle B}\in {\dot{\mathbf{H}}}^4(D)\), such that
i.e., \(T_{\scriptscriptstyle B}f:=v_{\scriptscriptstyle B}\). It is well known that, for \(m\in {\mathbb {N}}_0\), there exists a constant \(C^{m}_{\scriptscriptstyle B}>0\), such that
Due to the type of boundary conditions of (2.10), we have
which, after using (2.8), yields
Let \(({\mathcal {S}}(t)w_0)_{\scriptscriptstyle t\in [0,T]}\) be the standard semigroup notation for the solution w to (1.4). Then [see Appendix A in Kossioris and Zouraris (2013)], for \(\ell \in {\mathbb {N}}_0\), \(\beta \ge 0\) and \(p\ge 0\), there exists a constant \({\mathcal {C}}_{\beta ,\ell ,\mu ,\mu ^2 T}>0\), such that
for all \(w_0\in {\dot{\mathbf{H}}}^{p+4\ell -2\beta -2}(D)\) and \(t_a\), \(t_b\in [0,T]\) with \(t_b>t_a\).
2.3 Discrete operators
Let \(r=2\) or 3, and \(\mathsf{M}_h^r\subset H_0^1(D)\cap H^2(D)\) be a finite element space consisting of functions which are piecewise polynomials of degree at most r over a partition of D in intervals with maximum length h. It is well known (cf., e.g., Bramble and Hilbert 1970) that
where \(C_{r}\) is a positive constant that depends on r and D, and is independent of h and v. Then, we define the discrete biharmonic operator \(B_h:\mathsf{M}_h^r\rightarrow \mathsf{M}_h^r\) by \((B_h\varphi ,\chi )_{\scriptscriptstyle 0,D}=(\partial _x^2\varphi ,\partial _x^2\chi )_{\scriptscriptstyle 0,D}\) for \(\varphi ,\chi \in \mathsf{M}_h^r\), the \(L^2(D)\)-projection operator \(P_h:L^2(D)\rightarrow \mathsf{M}_h^r\) by \((P_hf,\chi )_{\scriptscriptstyle 0,D}=(f,\chi )_{\scriptscriptstyle 0,D}\) for \(\chi \in \mathsf{M}_h^r \) and \(f\in L^2(D)\), and the standard Galerkin finite element approximation \(v_{{\scriptscriptstyle B},h}\in \mathsf{M}_h^r\) of the solution \(v_{\scriptscriptstyle B}\) to (2.10) by requiring
Let \(T_{{\scriptscriptstyle B},h}:L^2(D)\rightarrow {\mathsf{M}^r_h}\) be the solution operator of the finite element method (2.16), i.e., \(T_{{\scriptscriptstyle B},h}f:=v_{{\scriptscriptstyle B},h}=B_h^{-1}P_hf\) for all \(f\in L^2(D)\). Then, we can easily conclude that
and
Finally, the approximation property (2.15) of the finite element space \(\mathsf{M}_h^r\) yields (see, e.g., Proposition 2.2 in Kossioris and Zouraris 2010) the following error estimate:
3 An approximation estimate for the canvas problem solution
Here, we establish the convergence of \(\mathsf{u}\) towards u with respect to the \(L^{\infty }_t(L^2_{\scriptscriptstyle P}(L^2_x))\) norm, when \(\Delta {t}\rightarrow 0\) and \(\mathsf{M}\rightarrow \infty \) (cf. Kossioris and Zouraris 2010, 2013).
Theorem 3.1
Let u be the solution to (1.1), \(\mathsf{u}\) be the solution to (1.5), and \(\kappa \in {\mathbb {N}}\), such that \(\kappa ^2\,\pi ^2>\mu \). Then, there exists a constant \({\widehat{c}}_{\scriptscriptstyle \mathrm{CER}}>0\), independent of \(\Delta {t}\) and \(\mathsf{M}\), such that
where \({\mathsf{\Theta }}(t):=\left( {\mathbb {E}}\left[ \Vert u(t,\cdot )-\mathsf{u}(t,\cdot )\Vert _{\scriptscriptstyle 0,D}^2 \right] \right) ^{\frac{1}{2}}\) for \(t\in [0,T]\).
Proof
In the sequel, we will use the symbol C to denote a generic constant that is independent of \(\Delta {t}\) and \(\mathsf{M}\) and may change value from one line to the other.
Using (1.2), (1.9), and Lemma 2.1, we conclude that
for \((t,x)\in [0,T]\times {\overline{D}}\), where \({\widetilde{\mathsf{\Psi }}}: (0,T)\times {D}\rightarrow L^2({\mathcal {D}})\) is given by
for \((s,y)\in T_n\times {D}\), \(n=1,\dots ,\mathsf{N}\), and for \((t,x)\in (0,T]\times D\). Now, we use (1.3) and the \(L^2(D)\)-orthogonality of \((\varphi _k)_{k=1}^{\infty }\) to obtain
for \((s,y)\in T_n\times {D}\), \(n=1,\dots ,\mathsf{N}\), and for \((t,x)\in (0,T]\times D\). In addition, we use (3.2), (2.4), and (2.3), to get
where
and
Proceeding as in the proof of Theorem 3.1 in Kossioris and Zouraris (2013), we arrive at
Combining (3) and (3.3) and using the \(L^2(D)\)-orthogonality of \((\varepsilon _k)_{k=1}^{\infty }\) and \((\varphi _k)_{k=1}^{\infty }\), we have
For \(\mathsf{M}\ge \kappa \), using the Cauchy–Schwarz inequality, we obtain
The error bound (3.1) follows by observing that \(\Theta (0)=0\) and by combining the bounds (3.4), (3.5) and (3.6). \(\square \)
4 Deterministic time-discrete and fully discrete approximations
In this section, we define and analyze auxiliary time-discrete and fully discrete approximations of the solution to the deterministic problem (1.4). The results of the convergence analysis will be used in Sect. 5 for the derivation of an error estimate for the numerical approximations of \(\mathsf{u}\) introduced in Sect. 1.2.
4.1 Time-discrete approximations
We define an auxiliary modified-IMEX time-discrete method to approximate the solution w to (1.4), which has the following structure: First, sets
and determines \(W^1\in {\dot{\mathbf{H}}}^4(D)\) by
Then, for \(m=2,\dots ,M\), finds \(W^m\in {\dot{\mathbf{H}}}^4(D)\), such that
In the proposition below, we derive a low regularity priori error estimate in a discrete in time \(L^2_t(L^2_x)\)-norm.
Proposition 4.1
Let \((W^m)_{m=0}^{\scriptscriptstyle M}\) be the time-discrete approximations defined in (4.1)–(4.3), and w be the solution to the problem (1.4). Then, there exists a constant \(C>0\), independent of \(\Delta \tau \), such that
where \(w^{\ell }(\cdot ):=w(\tau _{\ell },\cdot )\) for \(\ell =0,\dots ,M\).
Proof
In the sequel, we will use the symbol C to denote a generic constant that is independent of \(\Delta \tau \) and may changes value from one line to the other.
Let \(\mathsf{E}^m:=w^m-W^m\) for \(m=0,\dots ,M\), and
for \(m=1,\dots ,M\). Thus, combining (1.4), (4.2) and (4.3), we conclude that
First, take the \(L^2(D)\)-inner product of both sides of (4.5) with \(\mathsf{E}^1\) and of (4.6) with \(\mathsf{E}^m\), and then use (2.13) to obtain
for \(m=2,\dots ,M\). Then, using that \(\mathsf{E}^0=0\) and applying (2.5) along with the arithmetic mean inequality, we get
Observing that (4.8) yields
we use a standard discrete Gronwall argument to arrive at
Summing both sides of (4.8) with respect to m, from 2 up to M, we obtain
which, along with (4.9), yields
Using (4.7), (2.8), the Cauchy–Schwarz inequality and the arithmetic mean inequality, we have
which, finally, yields
Next, we use the Cauchy–Schwarz inequality and (2.9) to get
Finally, we use (4.10), (4.11), (4.12), and (2.14) (with \(\beta =0\), \(\ell =1\), and \(p=0\)) to obtain
which establishes (4.4) for \(\theta =1\).
From (4.2), (4.3), and (2.12), it follows that:
Taking the \(L^2(D)\)-inner product of both sides of the first equation above with \(W^1\) and of the second one with \(W^m\), and then applying (2.13), (2.5) and the arithmetic mean inequality, we obtain
The inequalities (4.13) and (4.14), easily, yield that
from which, after the use of a standard discrete Gronwall argument, we arrive at
We sum both sides of (4.14) with respect to m, from 2 up to M, and then use (4.15), to have
Thus, using (4.16), (4.13), (4.1), (2.9), and (2.2), we obtain
In addition, we have
which, along with (2.14) (with \((\beta ,\ell ,p)=(0,0,0)\) and \((\beta ,\ell ,p)=(2,1,0)\)), yields
Thus, (4.17) and (4.18) establish (4.4) for \(\theta =0\).
Finally, the estimate (4.4) follows by interpolation, since it is valid for \(\theta =1\) and \(\theta =0\). \(\square \)
We close this section by deriving, for later use, the following a priori bound.
Lemma 4.1
Let \((W^m)_{m=0}^{\scriptscriptstyle M}\) be the time-discrete approximations defined by (4.1)–(4.3). Then, there exist a constant \(C>0\), independent of \(\Delta \tau \), such that
Proof
In the sequel, we will use the symbol C to denote a generic constant that is independent of \(\Delta \tau \) and may changes value from one line to the other.
Taking the \((\cdot ,\cdot )_{\scriptscriptstyle 0,D}\)-inner product of (4.3) with \(\partial _x^2W^m\) and of (4.2) with \(\partial _x^2W^1\), and then integrating by parts, we obtain
for \(m=2,\dots ,M\). Using (2.5) and the arithmetic mean inequality, from (4.20) and (4.21), it follows that:
Now, (4.23) and (4.22), easily, yield that
which, after a standard induction argument, leads to
After summing both sides of (4.23) with respect to m, from 2 up to M, we obtain
which, after using (4.24), yields
Finally, we combine (4.25), (4.22), and (2.1) to get
which, easily, yields (4.19). \(\square \)
4.2 Fully discrete approximations
The modified-IMEX time-stepping method along with a finite element space discretization yields a fully discrete method for the approximation of the solution to the deterministic problem (1.4). The method begins by setting
and specifying \(W_h^1\in \mathsf{M}_h^r\), such that
Then, for \(m=2,\dots ,M\), it finds \(W_h^m\in \mathsf{M}_h^r\), such that
Adopting the viewpoint that the fully discrete approximations defined above are approximations of the time-discrete ones defined in the previous section, we estimate below the corresponding approximation error in a discrete in time \(L^2_t(L^2_x)\)-norm.
Proposition 4.2
Let \(r=2\) or 3, \((W^m)_{m=0}^{\scriptscriptstyle M}\) be the time-discrete approximations defined by (4.1)–(4.3), and \((W_h^m)_{m=0}^{\scriptscriptstyle M}\subset \mathsf{M}_h^r\) be the fully discrete approximations specified in (4.26)–(4.28). Then, there exists a constant \(C>0\), independent of \(\Delta \tau \) and h, such that
Proof
In the sequel, we will use the symbol C to denote a generic constant which is independent of \(\Delta \tau \) and h, and may changes value from one line to the other.
Let \(\mathsf{Z}^m:=W^m-W_h^m\) for \(m=0,\dots ,M\). Then, from (4.2), (4.3), (4.27), and (4.28), we obtain the following error equations:
where
Taking the \(L^2(D)\)-inner product of both sides of (4.31) with \(\mathsf{Z}^m\), we obtain
which, along with (2.17) and (2.5), yields
for \(m=2,\dots ,M\), where
Using (2.17), integration by parts, the Cauchy–Schwarz inequality, the arithmetic mean inequality, we have
and
Now, we combine (4.33), (4.34) and (4.35) to get
for \(m=2,\dots ,M\). Let \(\Upsilon ^{\ell }:=\Vert \partial _x^2(T_{\scriptscriptstyle B,h}\mathsf{Z}^{\ell })\Vert _{\scriptscriptstyle 0,D}^2+\Delta \tau \,\Vert \mathsf{Z}^{\ell }\Vert _{\scriptscriptstyle 0,D}^2\) for \(\ell =1,\dots ,M\). Then, (4.36) yields
from which, after applying a standard discrete Gronwall argument, we conclude that
Since \(T_{\scriptscriptstyle B,h}\mathsf{Z}^0=0\), after taking the \(L^2(D)\)-inner product of both sides of (4.30) with \(\mathsf{Z}^1\), and then, using (2.17) and the arithmetic mean inequality, we obtain
which, along with (4.37), yields
Now, summing both sides of (4.36) with respect to m, from 2 up to M, we obtain
which, along with (4.39), yields
Combining (4.40), (4.32), (2.19), and (4.19), we obtain
Thus, (4.41) yields (4.29) for \(\theta =1\).
From (4.27) and (4.28), we conclude that
Taking the \(L^2(D)\)-inner product of both sides of the first equation above with \(W_h^1\) and of the second one with \(W_h^m\), and then, applying (2.17) and (2.5), we obtain
where
Using (2.17), integration by parts, the Cauchy–Schwarz inequality, and the arithmetic mean inequality, we have
Combining (4.43) and (4.44), we arrive at
Let\(\Upsilon _h^{\ell }:=\Vert \partial _x^2(T_{\scriptscriptstyle B,h}W_h^{\ell })\Vert _{\scriptscriptstyle 0,D}^2 +\Delta \tau \,\Vert W_h^{\ell }\Vert _{\scriptscriptstyle 0,D}^2\) for \(\ell =1,\dots ,M\). Then, we use (4.42), (4.26), (2.18), (2.2), and (4.45) to obtain
and
From (4.47), after the application of a standard discrete Gronwall argument and the use of (4.46), we conclude that
Summing both sides of (4.45) with respect to m, from 2 up to M, we have
which, along with (4.48), yields
Thus, (4.49) and (4.17) yield (4.29) for \(\theta =0\).
Thus, the error estimate (4.29) follows by interpolation, since it holds for \(\theta =1\) and \(\theta =0\). \(\square \)
5 Convergence analysis of the IMEX finite element method
To estimate the approximation error of the IMEX finite element method given in Sect. 1.2, we use, as a tool, the corresponding IMEX time-discrete approximations of \(\mathsf{u}\), which are defined first by setting
and then, for \(m=1,\dots ,M\), by seeking \(\mathsf{U}^m\in {\dot{\mathbf{H}}}^4(D)\), such that
Thus, we split the total error of the IMEX finite element method as follows:
where \(\mathsf{u}^m:=\mathsf{u}(\tau _m,\cdot )\), \({\mathcal {E}}^m_{{\scriptscriptstyle \mathrm TDR}}:= \left( {\mathbb {E}}\left[ \Vert \mathsf{u}^m-\mathsf{U}^m\Vert _{\scriptscriptstyle 0,D}^2 \right] \right) ^{\scriptscriptstyle 1/2}\) is the time-discretization error at \(\tau _m\), and \({\mathcal {E}}_{{\scriptscriptstyle \mathrm SDR}}^m:= \left( {\mathbb {E}}\left[ \Vert \mathsf{U}^m-\mathsf{U}^m_h\Vert _{\scriptscriptstyle 0,D}^2 \right] \right) ^{\scriptscriptstyle 1/2}\) is the space-discretization error at \(\tau _m\).
5.1 Estimating the time-discretization error
The convergence estimate of Proposition 4.1 is the main tool in providing a discrete in time \(L^{\infty }_t(L^2_{\scriptscriptstyle P}(L^2_x))\) error estimate of the time-discretization error (cf. Yan 2005; Kossioris and Zouraris 2010, 2013).
Proposition 5.1
Let \(\mathsf{u}\) be the solution to (1.5) and \((\mathsf{U}^m)_{m=0}^{\scriptscriptstyle M}\) be the time-discrete approximations of \(\mathsf{u}\) defined by (5.1)–(5.2). Then, there exists a constant \({\widehat{c}}_{{\scriptscriptstyle \mathrm TDR}}\), independent of \(\Delta {t}\), \(\mathsf{M}\) and \(\Delta \tau \), such that
Proof
In the sequel, we will use the symbol C to denote a generic constant that is independent of \(\Delta {t}\), \(\mathsf{M}\), and \(\Delta \tau \), and may change value from one line to the other.
First, we introduce some notation by letting \(\mathsf{I}:L^2(D)\rightarrow L^2(D)\) be the identity operator, \(\mathsf{Y}:H^2(D)\rightarrow L^2(D)\) be the differential operator \(\mathsf{Y}:=\mathsf{I}-\Delta \tau \,\mu \,\partial _x^2\), and \(\mathsf{\Lambda }:L^2(D)\rightarrow {\dot{\mathbf{H}}}^4(D)\) be the inverse elliptic operator \(\mathsf{\Lambda }:=(\mathsf{I}+\Delta \tau \,\partial _x^4)^{-1}\). Then, for \(m=1,\dots ,M\), we define the operator \(\mathsf{Q}^m:L^2(D)\rightarrow {\dot{\mathbf{H}}}^4(D)\) by \(\mathsf{Q}^m:=(\mathsf{\Lambda }\circ \mathsf{Y})^{m-1}\circ \mathsf{\Lambda }\). In addition, for given \(w_0\in {\dot{\mathbf{H}}}^2(D)\), let \(({\mathcal {S}}_{\scriptscriptstyle {\Delta \tau }}^m(w_0))_{m=0}^{\scriptscriptstyle M}\) be time-discrete approximations of the solution to the deterministic problem (1.4), defined by (4.1)–(4.3). Then, using a simple induction argument, we conclude that
Let \(m\in \{1,\dots ,M\}\). Applying a simple induction argument on (5.2), we conclude that
which, along with (1.6) and (5.5), yields
In addition, using (1.9) and (1.6), and proceeding in similar manner, we arrive at
Thus, using (5.6) and (5.7) along with Remark 1.8, we obtain
which, easily, yields
with
Proceeding as in the proof of Theorem 4.1 in Kossioris and Zouraris (2013), we get
In addition, using the error estimate (4.4), it follows that:
Setting \(\theta =\tfrac{1}{8}-\epsilon \) with \(\epsilon \in \left( 0,\tfrac{1}{8}\right] \), we have
Thus, the estimate (5.4) follows, easily, as a simple consequence of (5.8), (5.9), and (5.10). \(\square \)
5.2 Estimating the space-discretization error
The outcome of Proposition 4.2 will be used below in the derivation of a discrete in time \(L^{\infty }_t(L^2_{\scriptscriptstyle P}(L^2_x))\) error estimate of the space-discretization error (cf. Yan 2005; Kossioris and Zouraris 2010, 2013).
Proposition 5.2
Let \(r=2\) or 3, \((\mathsf{U}_h^m)_{m=0}^{\scriptscriptstyle M}\) be the fully discrete approximations defined by (1.10)–(1.11) and \((\mathsf{U}^m)_{m=0}^{\scriptscriptstyle M}\) be the time-discrete approximations defined by (5.1)–(5.2). Then, there exists a constant \({\widehat{c}}_{{\scriptscriptstyle \mathrm SDR}}>0\), independent of \(\mathsf{M}\), \(\Delta {t}\), \(\Delta \tau \) and h, such that
Proof
In the sequel, we will use the symbol C to denote a generic constant that is independent of \(\Delta {t}\), \(\mathsf{M}\), \(\Delta \tau \), and h, and may change value from one line to the other.
Let us denote by \(\mathsf{I}:L^2(D)\rightarrow L^2(D)\) the identity operator, by \(\mathsf{Y}_h:\mathsf{M}_h^r\rightarrow \mathsf{M}_h^r\) the discrete differential operator \(\mathsf{Y}_h:=\mathsf{I}-\mu \,\Delta \tau \,(P_h\circ \partial _x^2)\), \(\mathsf{\Lambda }_h:L^2(D)\rightarrow \mathsf{M}^r_h\) be the inverse discrete elliptic operator \(\mathsf{\Lambda }_h:=(I+\Delta \tau \,B_h)^{-1}\circ P_h\). Then, for \(m=1,\dots ,M\), we define the auxiliary operator \(\mathsf{Q}_h^m:L^2(D)\rightarrow \mathsf{M}_h^r\) by \(\mathsf{Q}^m_h:=(\mathsf{\Lambda }_h\circ \mathsf{Y}_h)^{m-1}\circ \mathsf{\Lambda }_h\). In addition, for given \(w_0\in {\dot{\mathbf{H}}}^2(D)\), let \(({\mathcal {S}}_{h}^m(w_0))_{m=0}^{\scriptscriptstyle M}\) be fully discrete approximations of the solution to the deterministic problem (1.4), defined by (4.26)–(4.28). Then, using a simple induction argument, we conclude that
Let \(m\in \{1,\dots ,M\}\). Using a simple induction argument on (1.11), (1.6) and (5.12), we conclude that
After, using (5.13), (5.6), and Remark 1.8, and proceeding as in the proof of Proposition 5.1, we arrive at
which, along (4.29), yields
Setting \(\theta =\tfrac{1}{6}-\delta \) with \(\delta \in \left( 0,\tfrac{1}{6}\right] \), we have
which obviously yields (5.11) with \(\epsilon =r\delta \). \(\square \)
5.3 Estimating the total error
Theorem 5.3
Let \(r=2\) or 3, \(\mathsf{u}\) be the solution to the problem (1.5), and \((\mathsf{U}_h^m)_{m=0}^{\scriptscriptstyle M}\) be the finite element approximations of \(\mathsf{u}\) constructed by (1.10)–(1.11). Then, there exists a constant \({\widehat{c}}_{\scriptscriptstyle \mathrm{TTL}}>0\), independent of h, \(\Delta \tau \), \(\Delta {t}\) and \(\mathsf{M}\), such that
for all \(\epsilon _1\in \left( 0,\tfrac{1}{8}\right] \) and \(\epsilon _2\in \left( 0,\frac{r}{6}\right] \).
Proof
The error bound (5.15) follows easily from (5.4), (5.11), and (5.3). \(\square \)
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Acknowledgements
Work partially supported by The Research Committee of The University of Crete under Research Grant #4339: ‘Numerical solution of stochastic partial differential equations’ funded by The Research Account of the University of Crete (2015–2016).
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Communicated by Jorge Zubelli.
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Zouraris, G.E. An IMEX finite element method for a linearized Cahn–Hilliard–Cook equation driven by the space derivative of a space–time white noise. Comp. Appl. Math. 37, 5555–5575 (2018). https://doi.org/10.1007/s40314-018-0650-2
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DOI: https://doi.org/10.1007/s40314-018-0650-2
Keywords
- Finite element method
- Space derivative of a space–time white noise
- Spectral representation of the noise
- Implicit/explicit time-stepping
- Fully discrete approximations
- A priori error estimates