Finite element approximations of the stochastic mean curvature flow of planar curves of graphs

Abstract

This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for planar curves of graphs. To circumvent the difficulty caused by the low spatial regularity of the SPDE solution, a regularization procedure is first proposed to approximate the SPDE, and an error estimate for the regularized problem is derived. A semi-discrete finite element method, and a space-time fully discrete method are then proposed to approximate the solution of the regularized SPDE problem. We show $L^2$-convergence with rates for both, semi- and fully discrete 1methods. Computational experiments are provided to study the interplay of the geometric evolution and gradient type-noises.

1 Introduction

The mean curvature flow (MCF) refers to a one-parameter family of hypersurfaces $\{\Gamma _t\}_{t\ge 0} \subset \mathbf {R}^{d+1}$ which starts from a given initial surface $\Gamma _0$ and evolves according to the geometric law

$$\begin{aligned} V_n(t, \cdot ) = H(t, \cdot ), \end{aligned}$$

where $V_n(t, \cdot )$ and $H(t,\cdot )$ denote respectively the inward normal velocity and the mean curvature of the hypersurface $\Gamma _t$ at time $t$. The MCF is the best known curvature-driven geometric flow which finds many applications in differential geometry, geometric measure theory, image processing and materials science and have been extensively studied both analytically and numerically (cf. [11, 17, 27, 31, 34] and the references therein).

As a geometric problem, the MCF can be described using different formulations. Among them, we mention the classical parametric formulation [19], Brakke’s varifold formulation [2], De Giorgi’s barrier function formulation [3, 4, 10], the variational formulation [1], the level set formulation [8, 15, 28], and the phase field formulation [14, 20]. We remark that different formulations often lead to different solution concepts and also lead to developing different analytical (and numerical) concepts and techniques to analyze and approximate the MCF. However, all these formulations of the MCF give rise to difficult but interesting nonlinear geometric partial differential equations (PDEs), and the resolution of the MCF then depends on the solutions of these nonlinear geometric PDEs. One interesting feature of the MCF is the development of singularities, in particular singularities which may occur in finite time, even when the initial hypersurface is smooth. The singularities may appear in different forms such as self-intersection, pinch-off, merging, and fattening. To understand and characterize these singularities have been the focus of the analytical and numerical research on the MCF (cf. [8, 11, 12, 15, 16, 25–27, 31, 32, 34], and the references therein).

For application problems, there is a great deal of interest to include stochastic effects, and to study the impact of special noises on regularities of solutions, as well as their long-time behaviors. The uncertainty may arise from various sources such as thermal fluctuation, impurities of the materials, and the intrinsic instabilities of the deterministic evolutions. In this paper we consider the following form of a stochastically perturbed mean curvature flow:

$$\begin{aligned} V_n= H(t,\cdot ) + \epsilon \dot{W}_t, \end{aligned}$$

(1.1)

where $\dot{W}$ denotes a white in time noise, and $\epsilon >0$ is a constant. It is easy to check that (cf. [13, 33]) the level set formulation of (1.1) is given by the following nonlinear parabolic stochastic partial differential equation (SPDE):

$$\begin{aligned} df=|\nabla _{x'} f|\, \text{ div }_{x'} \left( \frac{\nabla _{x'} f}{|\nabla _{x'} f|} \right) \, dt + \epsilon |\nabla _{x'} f| \circ dW_t, \end{aligned}$$

(1.2)

where $f=f(x',t)$ with $x'=(x, x_{d+1})$ denotes the level set function so that $\Gamma _t$ is represented by the zero level set of $f$, and ‘$\circ$’ refers to the Stratonovich interpretation of the stochastic integral. Again, stochastic effects are modeled by a standard ${\mathbb R}$-valued Wiener process $W \equiv \{ W_t;\, t \ge 0\}$ which is defined on a given filtered probability space $(\Omega , {\mathcal F}, \{ {\mathcal F}_t;\, t \ge 0\}, {\mathbb P})$.

In the case that $f$ is a $d$-dimensional graph, that is, $f(x',t)=x_{d+1}- u(x,t)$, equation (1.2) reduces to

$$\begin{aligned} du=\sqrt{1+|\nabla _x u|^2}\, \text{ div }_{x} \left( \frac{\nabla _x u}{\sqrt{1+|\nabla _x u|^2}} \right) \, dt + \epsilon \sqrt{1+|\nabla _x u|^2} \circ dW_t. \end{aligned}$$

(1.3)

To the best of our knowledge, a comprehensive PDE theory for the SPDE (1.3) is still missing in the literature. For the case $d=1$, (1.3) reduces to the following one-dimensional nonlinear parabolic SPDE:

$$\begin{aligned} du&= \frac{\partial ^2_x u}{1+|\partial _x u|^2 } dt + \epsilon \sqrt{1+|\partial _x u|^2}\circ dW_t \nonumber \\&= \partial _x \bigl ( \mathrm{arctan} (\partial _x u)\bigr ) dt + \epsilon \sqrt{1+|\partial _x u|^2} \circ dW_t. \end{aligned}$$

(1.4)

Here $\partial _x u$ stands for the derivative of $u$ with respect to $x$. This Stratonovich SPDE can be equivalently converted into the following Itô SPDE:

$$\begin{aligned} du&= \left[ \frac{\epsilon ^2}{2} \partial ^2_x u +\left( 1-\frac{\epsilon ^2}{2}\right) \frac{\partial _x^2 u}{\sqrt{1+|\partial _x u|^2} } \right] dt + \epsilon \sqrt{1+|\partial _x u|^2}\, dW_t \nonumber \\&=\partial _x \left( \frac{\varepsilon ^2}{2} \partial _x u + \left( 1- \frac{\varepsilon ^2}{2}\right) \mathrm{arctan} (\partial _x u)\right) dt + \epsilon \sqrt{1+|\partial _x u|^2}\, dW_t. \end{aligned}$$

(1.5)

As is evident from (1.4), (1.5), the stochastic mean curvature flow (1.3) for $d=1$ may be interpreted as a gradient flow with multiplicative noise. Recently, Es-Sarhir and von Renesse [13] proved existence and uniqueness of (stochastically) strong solutions for (1.4) by a variational method, based on the Lyapunov structure of the problem $\{$cf. [13, property (H3)]$\}$ which replaces the standard coercivity assumption $\{$cf.  [13, property (A)]$\}$. As is pointed out in [13], mild solutions for (1.4) may not be expected due to its quasilinear character.

The primary goal of this paper is to develop and analyze by a variational method some semi-discrete and fully discrete finite element methods for approximating (with rates) the (stochastically) strong solution of the Itô form (1.5) of the stochastic MCF. The error analysis presented in this paper differs from most existing works on the numerical analysis of SPDEs, where mild solutions are mostly approximated with the help of corresponding discrete semi-groups (see  [22] and the references therein). We also note that the error estimates derived in [18] which hold for general quasilinear SPDEs do not apply to (1.5) because the structural assumptions, such as the coercivity assumption [18, cf. Assumption 2.1, (ii)] and the strong monotonicity assumption [18, cf. Assumption 2.2, (i)] fail to hold for (1.5), and also the regularity assumptions [18, cf. Assumption 2.3] are not known to hold in the present case. In this paper, we use a variational approach similar to [6, 7, 18, 24, 29, 30] to analyze the convergence of our finite element methods. One main difficulty for approximating the strong solution of (1.5) with certain rates is caused by the low regularity of the solution. To circumvent this difficulty, we first regularize the SPDE (1.5) by adding an additional linear diffusion term $\delta \partial _x^2 u$ to the drift coefficient of (1.5); as a consequence the related drift operator in (2.3) becomes strongly monotone, and the corresponding solution process $u^\delta$ is then $H^2$-valued in space. However, it is due to the ‘gradient-type’ noise that a relevant Hölder estimate in the $H^1$-norm for the solution $u^\delta$ seems not available, which is necessary to properly control time-discretization errors. In order to circumvent this problematic issue, we proceed first with the spatial discretization (3.1): we may then use an inverse finite element estimate, and the weaker Hölder estimate (3.16) for the process $u^\delta _h$ to control time-discretization errors. We remark that addressing space discretization errors first requires to efficiently cope with the limited regularity of Lagrange finite element functions in the context of required higher norm estimates, which is overcome by a perturbation argument (cf. Proposition 3.2).

The remainder of this paper consists of three additional sections. In Sect. 2 we first recall some relevant facts about the solution of (1.5) from [13]; we then present an analysis for the regularized problem. The main result of this section is to prove an error bound for $u^\delta -u$ in powers of $\delta$. In Sect. 3 we propose a semi-discrete (in space) and a fully discrete finite element method for the regularized Eq. (2.3) of the SPDE (1.5). The main result of this section is the $L^2$-error estimate for the finite element solution. Finally, in Sect. 4 we present several computational results to validate the theoretical error estimate, and to study relative effects due to geometric evolution and gradient-type noises.

2 Preliminaries and error estimates for a PDE regularization

The standard function and space notation will be adapted in this paper. For example, $H^2(I)$ denotes the Sobolev space $W^{2,2}(I)$ on the interval $I=(0,1)$, and $H^0(I)=L^2(I)$. We also use $H^m_p(I)$ to denote the subspace of $H^m(I)$ which consists of all periodic functions in $H^m(I)$. Let $(\cdot , \cdot )_I$ denote the $L^2$-inner product on $I$. The quadruple $(\Omega , \mathcal {F}, \{ {\mathcal F}_t; t \ge 0\}, {\mathbb P})$ stands for a given filtered probability space, on which an ${\mathbb R}$-valued Wiener process $W$ is given. For a random variable $X$, we denote by $\mathbb {E}[X]$ the expected value of $X$.

We first quote the following existence and uniqueness result from [13] for the SPDE (1.5) with periodic boundary conditions. In this context, we refer to the $\{ {\mathcal F}_t\}$-adapted process $u: I \times [0,T] \times \Omega \rightarrow {\mathbb R}$ as a (stochastically) strong solution in case it satisfies ${\mathbb P}$-a.s. (1.5) in an analytically weak sense, i.e., tested with deterministic functions.

Theorem 2.1

Suppose that $u_0\in H^1_p(I)$ and fix $T > 0$. Let $\epsilon \le \sqrt{2}$. There exists a unique strong solution to SPDE (1.4) with periodic boundary conditions and attaining the initial condition $u(0)= u_0$, that is, there exists a unique $H^1_p$-valued $\{ {\mathcal F}_t\}_{t \in [0,T]}$-adapted process $u \equiv \{ u(t);\, t \in [0,T]\}$ such that ${\mathbb P}$-almost surely

$$\begin{aligned} \quad \bigl (u(t), \varphi \bigr )_I&= \bigl ( u_0,\varphi \bigr )_I -\frac{\epsilon ^2}{2} \int \limits _0^t \bigl ( \partial _x u, \partial _x \varphi \bigr )_I\, ds \\&-\left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _0^t \bigl ( \arctan (\partial _x u), \partial _x \varphi \bigr )_I \Bigr ] ds \nonumber \\&+\, \epsilon \int \limits _0^t \Bigl ( \sqrt{1+|\partial _x u|^2}, \varphi \Bigr )_I dW_s \quad \forall \varphi \in H^1_p(I) \quad \forall t\in [0,T].\nonumber \end{aligned}$$

(2.1)

Moreover, $u$ satisfies for some $C > 0$ independent of $T > 0$,

$$\begin{aligned} \sup _{t \in [0,T]} \mathbb {E} \bigl [\Vert u(t)\Vert _{H^1(I)}^2\bigr ] \le C. \end{aligned}$$

(2.2)

It is not clear if such a regularity can be improved from the analysis of [13] because of the difficulty caused by the gradient-type noise. In particular, $H^2$-regularity in space, which would be desirable in order to derive some rates of convergence for finite element methods, seems not clear. To overcome this difficulty, we introduce the following simple regularization of (1.5):

$$\begin{aligned} du^\delta =\left[ \left( \delta +\frac{\epsilon ^2}{2} \right) \partial _x^2 u^\delta +\left( 1-\frac{\epsilon ^2}{2}\right) \frac{\partial _x^2 u^\delta }{\sqrt{1+|\partial _x u^\delta |^2} } \right] dt + \epsilon \sqrt{1+|\partial _x u^\delta |^2}\, dW_t.\nonumber \\ \end{aligned}$$

(2.3)

To make this indirect approach successful, we need to address the well-posedness and regularity issues for (2.3) and to estimate the difference between the strong solutions $u^\delta$ of (2.3) and $u$ of (1.5).

Theorem 2.2

Suppose that $u^\delta _0\in H^1_p(I)$ and $\Vert u^\delta _0\Vert _{H^1(I)}\le C_0$, where $C_0 > 0$ is independent of $\delta$. Let $\epsilon \le \sqrt{2(1+\delta )}$. Then there exists a unique strong solution to SPDE (2.3) with periodic boundary conditions and initial condition $u^\delta (0)= u^\delta _0$, that is, there exists a unique $H^1_p$-valued $\{ {\mathcal F}_t\}_{t \in [0,T]}$-adapted process $u^\delta \equiv \{ u^\delta (t);\, t \in [0,T]\}$ such that there holds ${\mathbb P}$-almost surely

$$\begin{aligned} \bigl (u^\delta (t), \varphi \bigr )_I =&\bigl ( u^\delta _0,\varphi \bigr )_I -\left( \delta +\frac{\epsilon ^2}{2}\right) \int \limits _0^t \bigl ( \partial _x u^\delta , \partial _x \varphi \bigr )_I \, ds \nonumber \\&-\left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _0^t \bigl ( \arctan (\partial _x u^\delta ), \partial _x \varphi \bigr )_I \,ds \nonumber \\&+\epsilon \int \limits _0^t \Bigl ( \sqrt{1+|\partial _x u^\delta |^2}, \varphi \Bigr )_I dW_s \quad \forall \varphi \in H^1_p(I) \quad \forall \, t \in [0,T]. \end{aligned}$$

(2.4)

Moreover, $u^\delta$ satisfies

$$\begin{aligned} \sup _{t \in [0,T]} \mathbb {E} \Bigl [\frac{1}{2}\Vert \partial _x u^\delta (t)\Vert _{L^2(I)}^2 \Bigr ]&+\delta \, \mathbb {E} \left[ \int \limits _0^T \Vert \partial _x^2 u^\delta (s)\Vert _{L^2(I)}^2\, ds\right] \nonumber \\&\quad \le \mathbb {E} \Bigl [\frac{1}{2} \Vert \partial _x u^\delta _0\Vert _{L^2(I)}^2 \Bigr ]\le C(C_0). \end{aligned}$$

(2.5)

Proof

Existence of $u^{\delta }$ can be shown in the same way as done in Theorem 2.1 (cf.  [13]). To verify (2.5), we proceed formally and apply Ito’s formula (cf. e.g. [23], or [9, p. 105]) with $f(\cdot ) = \frac{1}{2} \Vert \partial _x \cdot \Vert ^2_{L^2(I)}$ to (a Galerkin approximation of) the solution $u^{\delta }$, and use integration by parts to get

$$\begin{aligned}&\frac{1}{2} \Vert \partial _x u^{\delta }(t)\Vert ^2_{L^2(I)} + \int \limits _0^t \left[ \left( \frac{\varepsilon ^2}{2} + \delta \right) \Vert \partial _x^2 u^{\delta }\Vert ^2_{L^2(I)} + \left( 1- \frac{\varepsilon ^2}{2}\right) \Bigl \Vert \frac{\partial _x^2 u^{\delta }}{\sqrt{1+ \vert \partial _x u^{\delta }\vert ^2}}\Bigr \Vert _{L^2}^2\right] \, ds \\&\quad = \frac{1}{2} \Vert \partial _x u^{\delta }_0\Vert ^2_{L^2(I)} + \frac{\varepsilon ^2}{2} \int \limits _{0}^t \Bigl \Vert \partial _x \sqrt{1 + \vert \partial _x u^{\delta }\vert ^2} \Bigr \Vert ^2_{L^2} \, ds + M_t \\&\quad = \frac{1}{2} \Vert \partial _x u^{\delta }_0\Vert ^2_{L^2(I)} + \frac{\varepsilon ^2}{2} \int \limits _{0}^t \Bigl \Vert \frac{\partial _x u^{\delta } \cdot \partial _x^2 u^{\delta }}{\sqrt{1 + \vert \partial _x u^{\delta }\vert ^2}} \Bigr \Vert _{L^2(I)}^2\, ds + M_t\qquad \forall \, t \in [0,T], \end{aligned}$$

where

$$\begin{aligned} M_t:= \epsilon \int \limits _0^t \Bigl ( \partial _x \sqrt{1 + \vert \partial _x u^{\delta }(s)\vert ^2}, \partial _x u^\delta \Bigr )_I\, dW_s \end{aligned}$$

is a martingale. Taking expectation yields

$$\begin{aligned}&{\mathbb E} \left[ \frac{1}{2} \Vert \partial _x u^{\delta }(t)\Vert ^2_{L^2(I)} + \int \limits _0^t \left[ \delta \Vert \partial _x^2 u^{\delta }\Vert ^2_{L^2} + \left( 1- \frac{\varepsilon ^2}{2} \right) \Bigl \Vert \frac{\partial _x^2 u^{\delta }}{\sqrt{1+ \vert \partial _x u^{\delta }\vert ^2}}\Bigr \Vert _{L^2}^2 \right] \, ds \right] \\&\quad \le {\mathbb E}\left[ \frac{1}{2} \Vert \partial _x u^{\delta }_0\Vert ^2_{L^2}\right] \!. \end{aligned}$$

Hence, (2.5) holds. The proof is complete.$\square$

Next, we shall derive an upper bound for the error $u^\delta -u$ as a low order power function of $\delta$.

Theorem 2.3

Suppose that $u^\delta _0\equiv u_0$. Let $u$ and $u^\delta$ denote respectively the strong solutions of the initial-boundary value problems (1.5) and (2.3) as stated in Theorems 2.1 and 2.2. Then there holds the following error estimate:

$$\begin{aligned} \sup _{t\in [0,T]} \mathbb {E} \left[ \Vert u^\delta (t)-u(t)\Vert _{L^2(I)}^2 \right] + \delta \, \mathbb {E}\left[ \int \limits _0^T \Vert \partial _x \left( u^\delta (s)-u(s) \right) \Vert _{L^2(I)}^2\, ds \right] \le CT \delta . \end{aligned}$$

(2.6)

Proof

Let $e^\delta := u^\delta - u$. Subtracting (2.1) from (2.4) we get that ${\mathbb P}$-a.s.

$$\begin{aligned} \bigl (e^\delta (t), \varphi \bigr )_I =&- \int \limits _0^t \Bigl [ \delta \bigl ( \partial _x u, \partial _x \varphi \bigr )_I +\left( \delta +\frac{\epsilon ^2}{2} \right) \bigl ( \partial _x e^\delta , \partial _x \varphi \bigr )_I \\&+\left( 1-\frac{\epsilon ^2}{2}\right) \left( \arctan (\partial _x u^\delta ) -\arctan (\partial _x u), \partial _x \varphi \right) _I \Bigr ] ds + M_t \end{aligned}$$

for all $\varphi \in H^1(I)$ and $t \in [0,T]$, with the martingale

$$\begin{aligned} M_t:= \epsilon \int \limits _0^t \bigl ( \sqrt{1+|\partial _x u^\delta |^2}-\sqrt{1+|\partial _x u|^2}, \varphi \bigr )_I dW_s. \end{aligned}$$

By Itô’s formula (cf. [23]) we get

$$\begin{aligned} \Vert e^\delta (t)\Vert _{L^2(I)}^2&= - 2 \int \limits _0^t \left[ \delta \bigl ( \partial _x u, \partial _x e^\delta \right) _I +\bigl (\delta +\frac{\epsilon ^2}{2} \bigr ) \Vert \partial _x e^\delta \bigr \Vert ^2_{L^2(I)} \nonumber \\&\quad +\left( 1-\frac{\epsilon ^2}{2}\right) \bigl ( \arctan (\partial _x u^\delta ) -\arctan (\partial _x u), \partial _x e^\delta \bigr )_I \Bigr ] ds \nonumber \\&\quad + \epsilon ^2 \int \limits _0^t \Bigl \Vert \sqrt{1+|\partial _x u^\delta |^2}-\sqrt{1+|\partial _x u|^2} \Bigr \Vert _{L^2(I)}^2\, ds \nonumber \\&\quad + 2\epsilon \int \limits _0^t \left( \sqrt{1+|\partial _x u^\delta |^2}-\sqrt{1+|\partial _x u|^2}, e^\delta \right) _I dW_s. \end{aligned}$$

(2.7)

Taking expectations on both sides, and using the monotonicity property of the $\arctan$ function and the inequality $\bigl ( \sqrt{1+x^2} - \sqrt{1+y^2}\bigr )^2 \le \vert x - y\vert ^2$ yield

$$\begin{aligned}&\mathbb {E} \left[ \Vert e^\delta (t)\Vert _{L^2(I)}^2 \right] + 2 \delta \, \mathbb {E} \left[ \int \limits _0^t \Vert \partial _x e^\delta \Vert _{L^2(I)}^2 \, ds\right] \le - 2 \delta \, \mathbb {E} \left[ \int \limits _0^t \bigl ( \partial _x u, \partial _x e^\delta \bigr )_I ds\right] \\&\quad \le \delta \, \mathbb {E} \left[ \int \limits _0^T \left[ \Vert \partial _x u\Vert _{L^2(I)}^2 + \Vert \partial _x e^\delta \Vert _{L^2(I)}^2 \right] \, ds\right] \!, \end{aligned}$$

which together with (2.2) implies that

$$\begin{aligned} \mathbb {E} \left[ \Vert e^\delta (t)\Vert _{L^2(I)}^2\right] + \delta \, \mathbb {E} \left[ \int \limits _0^t \Vert \partial _x e^\delta \Vert _{L^2(I)}^2 \, ds\right]&\le \delta \, \mathbb {E} \left[ \int \limits _0^T \Vert \partial _x u\Vert _{L^2(I)}^2 \, ds\right] \\&\le (CT) \delta . \end{aligned}$$

The desired estimate (2.6) follows immediately. The proof is complete.$\square$

3 Finite element methods

In this section we propose a fully discrete finite element method to solve the regularized SPDE (2.3) and to derive an error estimate for the finite element solution. This goal will be achieved in two steps. We first present and study a semi-discrete in space finite element method and then discretize it in time to obtain our fully discrete finite element method.

3.1 Semi-discretization in space

Let $0=x_0<x_1<\cdots <x_{J+1}=1$ be a quasiuniform partition of $I=(0,1)$. Define $h_j:=x_{j+1}-x_j$ and $h:=\max _{0\le j\le J} h_j$. Introduce the finite element spaces

$$\begin{aligned} V^h_r:=\bigl \{v_h;\, v_h|_{[x_j,x_{j+1}]} \in P_r([x_j, x_{j+1}]),\, j=0,1,\ldots , J\bigr \} \cap H^1_p(I), \end{aligned}$$

where $P_r([x_j, x_{j+1}])$ denotes the space of all polynomials of degree not exceeding $r (\ge 0$) on $[x_j,x_{j+1}]$. We note that functions in $V^h_r$ are piecewise continuous periodic functions. Our semi-discrete finite element method for SPDE (2.3) is defined by seeking $u_h^\delta : [0,T]\times \Omega \rightarrow V^h_r$ such that ${\mathbb P}$-almost surely

$$\begin{aligned} \bigl (u_h^\delta (t), v_h \bigr )_I&= \bigl ( u_h^\delta (0), v_h \bigr )_I - \left( \delta +\frac{\epsilon ^2}{2}\right) \int \limits _0^t \bigl ( \partial _x u_h^\delta (s), \partial _x v_h \bigr )_I \, ds \nonumber \\&\quad -\left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _0^t \bigl ( \arctan (\partial _x u_h^\delta (s)), \partial _x v_h \bigr )_I \, ds \nonumber \\&\quad + \epsilon \int \limits _0^t \left( \sqrt{1+|\partial _x u_h^\delta (s)|^2}, v_h \right) _I dW_s \quad \forall v_h\in V^h_r \quad \forall t \in [0,T], \end{aligned}$$

(3.1)

where $u_h^\delta (0)=P_h^r u_0^\delta$, and $P_h^r$ denotes the $L^2$-projection operator from $L^2(I)$ to $V_r^h$ which is defined by

$$\begin{aligned} (P_h^r w, v_h)_I = (w,v_h)_I \quad \forall v_h \in V^h_r. \end{aligned}$$

To derive a SDE for $u_h^\delta$ from the above weak formulation, we introduce the discrete (nonlinear) operator $A_h^\delta : V_r^h\rightarrow V_r^h$ by

$$\begin{aligned} \bigl ( A_h^\delta w_h, v_h \bigr )_I :=&\left( \delta +\frac{\epsilon ^2}{2}\right) \bigl ( \partial _x w_h, \partial _x v_h \bigr )_I \\&+\left( 1-\frac{\epsilon ^2}{2}\right) \bigl ( \arctan (\partial _x w_h), \partial _x v_h \bigr )_I \quad \forall w_h, v_h\in V_r^h. \nonumber \end{aligned}$$

(3.2)

Then (3.1) can be equivalently written as

$$\begin{aligned} d u_h^\delta (t) = - A_h^\delta u_h^\delta (t) \, dt + \epsilon P_h\Bigl ( \sqrt{1+|\partial _x u_h^\delta (t)|^2} \Bigr )\, dW_t. \end{aligned}$$

(3.3)

Proposition 3.1

For $\epsilon \le \sqrt{2(1+\delta )}$, there is a unique solution $u_h^{\delta } \in C \bigl ( [0,T]; L^2(\Omega ;V_r^h)\bigr )$ to scheme (3.1). Moreover, there holds

$$\begin{aligned}&\sup _{0\le t\le T} \mathbb {E} \left[ \frac{1}{2} \bigl \Vert u_h^\delta (t) \bigr \Vert _{L^2(I)}^2 \right] + \delta \, \mathbb {E} \left[ \int \limits _0^T \bigl \Vert \partial _x u_h^\delta (s) \bigr \Vert _{L^2(I)}^2\, ds \right] \nonumber \\&\quad \le \mathbb {E} \left[ \frac{1}{2} \bigl \Vert u_h^{\delta }(0) \bigr \Vert _{L^2(I)}^2 \right] + \epsilon ^2 T. \end{aligned}$$

(3.4)

Proof

Well-posedness of (3.3) follows from the standard theory for stochastic ODEs with Lipschitz drift and diffusion. To verify (3.4), applying Itô’s formula (cf. [23], or [21, p. 92]) to $f(u_h^\delta )=\Vert u_h^\delta \Vert _{L^2(I)}^2$ and using (3.3) lead to

$$\begin{aligned} \Vert u_h^\delta (t)\Vert _{L^2(I)}^2 =&\, \Vert u_h^\delta (0)\Vert _{L^2(I)}^2 -2 \int \limits _0^t \Bigl ( A_h^\delta u_h^\delta (s), u_h^\delta (s) \Bigr )_I \, ds \\&+\epsilon ^2 \int \limits _0^t \Bigl \Vert P_h^r \sqrt{1+|\partial _x u_h^\delta (s)|^2 } \Bigr \Vert _{L^2(I)}^2 \, ds \nonumber \\&+2\epsilon \int \limits _0^t \Bigl ( P_h^r\sqrt{1+|\partial _x u_h^\delta (s)|^2}, u_h^\delta \Bigr )_I\, dW_s . \nonumber \end{aligned}$$

(3.5)

It follows from the definitions of $A_h^{\delta }$ and $P_h^r$ that

$$\begin{aligned} \Vert u_h^\delta (t)\Vert _{L^2(I)}^2 \le \Vert u_h^\delta (0)\Vert _{L^2(I)}^2&-(2\delta +\epsilon ^2)\int \limits _0^t \bigl \Vert \partial _x u_h^\delta (s) \bigr \Vert _{L^2(I)}^2\, ds \\&- (2-\epsilon ^2) \int \limits _0^t \Bigl ( \arctan (\partial _x u_h^\delta (s)), \partial _x u_h^\delta (s) \Bigr )_I \, ds \nonumber \\&+ \epsilon ^2 \int \limits _0^t \left[ 1+ \bigl \Vert \partial _x u_h^\delta (s) \bigr \Vert _{L^2(I)}^2 \right] \, ds \nonumber \\&+2\epsilon \int \limits _0^t \left( \sqrt{1+|\partial _x u_h^\delta (s)|^2}, u_h^\delta \right) _I\, dW_s. \nonumber \end{aligned}$$

(3.6)

Then (3.4) follows from applying expectation to (3.6), and using the coercivity of arctan. The proof is complete.$\square$

An a priori estimate for $u_h^\delta$ in stronger norms is more difficult to obtain, which is due to low global smoothness and local nature of finite element functions. We shall derive some of these estimates in Proposition 3.2 using a perturbation argument after establishing error estimates for $u_h^\delta$.

To derive error estimates for $u_h^\delta$, we introduce the elliptic $H^1$-projection $R_h^r: H^1(I)\rightarrow V_r^h$, i.e., for any $w \in H^1(I)$, $R_h^r w \in V_r^h$ is defined by

$$\begin{aligned} \bigl ( \partial _x [R_h^r w - w], \partial _x v_h \bigr )_I + \bigl (R_h^r w -w, v_h \bigr )_I = 0 \quad \forall v_h\in V_r^h. \end{aligned}$$

(3.7)

The following error bounds are well-known (cf. [5]),

$$\begin{aligned} \bigl \Vert w- R_h^r w \bigr \Vert _{L^2(I)} + h \bigl \Vert w -R_h^r w \bigr \Vert _{H^1(I)} \le C h^2 \Vert w\Vert _{H^2(I)} \quad \forall w\in H^2_p(I). \end{aligned}$$

(3.8)

Theorem 3.1

Let $\epsilon \le \sqrt{2(1+\delta )}$ and $r=1$. Then there holds

$$\begin{aligned}&\sup _{t\in [0,T]} \mathbb {E} \left[ \bigl \Vert u^\delta (t) - u_h^\delta (t) \bigr \Vert _{L^2(I)}^2\right] +\delta \, \mathbb {E} \left[ \int \limits _0^T \bigl \Vert \partial _x [u^\delta (s) - u_h^\delta (s)] \bigr \Vert _{L^2(I)}^2 \, ds \right] \nonumber \\&\quad \le C h^2 \bigl ( 1+ \delta ^{-2} \bigr ). \end{aligned}$$

(3.9)

Proof

Let

$$\begin{aligned} e^{\delta }(t):=u^\delta (t)-u_h^{\delta }(t) ,\quad \eta ^{\delta }: = u^\delta (t)-R_h^r u^{\delta }(t),\quad \xi ^{\delta }:= R_h^r u^{\delta }(t)-u_h^{\delta }(t). \end{aligned}$$

Then $e^{\delta }=\eta ^{\delta } + \xi ^{\delta }$. Subtracting (3.1) from (2.4) we obtain the following error equation which holds ${\mathbb P}$-almost surely:

$$\begin{aligned}&\left( e^\delta (t), v_h \right) _I + \left( \delta +\frac{\epsilon ^2}{2}\right) \int \limits _0^t \bigl ( \partial _x e^\delta (s), \partial _x v_h \bigr )_I ds \nonumber \\&\quad = - \left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _{0}^{t} \Bigl ( \arctan (\partial _x u^\delta (s)) -\arctan (\partial _x u_h^\delta (s)), \partial _x v_h \Bigr )_I ds \nonumber \\&\qquad + \epsilon \int \limits _{0}^{t} \Bigl ( \sqrt{1+|\partial _x u^\delta (s)|^2}-\sqrt{1+|\partial _x u_h^\delta (s)|^2}, v_h\Bigr )_I dW_s + \bigl ( e^\delta (0), v_h \bigr )_I \end{aligned}$$

(3.10)

for all $v_h\in V_r^h$. Substituting $e^\delta =\eta ^\delta +\xi ^\delta$ and rearranging terms leads to

$$\begin{aligned}&\bigl ( \xi ^\delta (t), v_h \bigr )_I + \left( \delta +\frac{\epsilon ^2}{2}\right) \int \limits _0^t \bigl ( \partial _x \xi ^\delta (s), \partial _x v_h \bigr )_I ds \nonumber \\&\qquad + \left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _{0}^{t} \Bigl ( \arctan \bigl (\partial _x u^\delta (s)\bigr ) -\arctan \bigl (\partial _x u_h^\delta (s)\bigr ), \partial _x v_h \Bigr )_I ds \nonumber \\&\quad = \epsilon \int \limits _{0}^{t} \Bigl ( \sqrt{1+|\partial _x u^\delta (s)|^2}-\sqrt{1+|\partial _x u_h^\delta (s)|^2}, v_h\Bigr )_I dW_s \nonumber \\&\qquad -\left( \delta +\frac{\epsilon ^2}{2} \right) \int \limits _0^t \bigl (\partial _x\eta ^\delta (s),\partial _x v_h\bigr )_I\, ds -\bigl ( \eta ^{\delta }(t), v_h \bigr )_I + \bigl ( e^\delta (0), v_h \bigr )_I. \end{aligned}$$

(3.11)

Applying Itô’s formula (cf. [23]) with $f(\xi ^\delta )=\Vert \xi ^\delta \Vert _{L^2(I)}^2$, and using (3.11) and (3.2) we obtain

$$\begin{aligned}&\bigl \Vert \xi ^{\delta }(t) \bigr \Vert _{L^2(I)}^2 + \bigl (2\delta +\epsilon ^2\bigr ) \int \limits _0^t \bigl \Vert \partial _x \xi ^\delta (s) \bigr \Vert _{L^2(I)}^2\, ds\nonumber \\&\qquad + (2-\epsilon ^2) \int \limits _0^t \Bigl ( \mathrm{arctan} \bigl (\partial _x R_h^r u^{\delta }(s) \bigr ) - \mathrm{arctan} \bigl (\partial _x u_h^{\delta }(s)\bigr ), \partial _x \xi ^{\delta }(s) \Bigr )_I ds \nonumber \\&\quad = -\bigl ( 2-\epsilon ^2\bigr ) \int \limits _0^t \Bigl ( \arctan \bigl (\partial _x u^\delta (s) \bigr )-\arctan \bigl (\partial _x R^r_h u^{\delta }(s)\bigr ), \partial _x \xi ^{\delta }(s)\Bigr )_I ds \nonumber \\&\qquad + \epsilon ^2 \int \limits _0^t \Bigl \Vert \sqrt{1+|\partial _x u^\delta (s)|^2}-\sqrt{1+|\partial _x u_h^\delta (s)|^2} \Bigr \Vert _{L^2(I)}^2\, ds \nonumber \\&\qquad +2 \epsilon \int \limits _0^t \Bigl ( \sqrt{1+|\partial _x u^\delta (s)|^2}-\sqrt{1+|\partial _x u_h^\delta (s)|^2}, \xi ^\delta (s) \Bigr )_I dW_s \nonumber \\&\qquad -\bigl (2\delta +\epsilon ^2\bigr ) \int \limits _0^t \bigl (\partial _x\eta ^\delta (s),\partial _x\xi _h(s)\bigr )_I\, ds -2\bigl ( \eta ^{\delta }(t), \xi ^{\delta }(t) \bigr )_I \nonumber \\&\qquad +2\bigl ( \eta ^\delta (0), \xi ^\delta (t) \bigr )_I + \bigl ( \xi ^\delta (0), \xi ^\delta (0) \bigr )_I. \end{aligned}$$

(3.12)

By the monotonicity of arctan, (3.8), (2.5), and the inequality $\bigl ( \sqrt{1+x^2} - \sqrt{1+y^2}\bigr )^2 \le \vert x - y\vert ^2$ we have

$$\begin{aligned}&{\mathbb E} \left[ \int \limits _0^t \Bigl ( \arctan (\partial _x R_h^r u^\delta (s))-\arctan (\partial _x u_h^{\delta }(s)), \partial _x \xi ^{\delta }(s)\Bigr )_I ds \right] \ge 0,\\&\quad (2-\epsilon ^2) {\mathbb E}\left[ \int \limits _0^t \Bigl ( \arctan \bigl (\partial _x u^\delta (s) \bigr )-\arctan \bigl (\partial _x R^r_h u^{\delta }(s)\bigr ), \partial _x \xi ^{\delta }(s)\Bigr )_I ds\right] \\&\qquad \le {\mathbb E}\left[ \int \limits _0^t \Bigl ( \frac{\delta }{4} \Vert \partial _x \xi ^\delta (s)\Vert _{L^2(I)}^2 + 4\delta ^{-1} \bigl \Vert \partial _x u^\delta (s)-\partial _x R_h^r u^\delta (s) \bigr \Vert _{L^2(I)}^2 \Bigr )\, ds\right] \\&\qquad \le \frac{\delta }{4} {\mathbb E}\Bigl [\int \limits _0^t \Vert \partial _x \xi ^\delta (s)\Vert _{L^2(I)}^2\, ds\Bigr ] + Ch^2 \delta ^{-2}, \\&{\mathbb E}\left[ \epsilon ^2 \int \limits _0^t \Bigl \Vert \sqrt{1+|\partial _x u^\delta (s)|^2}-\sqrt{1+|\partial _x u_h^\delta (s)|^2} \Bigr \Vert _{L^2(I)}^2\, ds\right] \\&\qquad \le {\mathbb E}\left[ \left( \epsilon ^2 + \frac{\delta }{4} \right) \int \limits _0^t \Vert \partial _x \xi ^\delta (s)\Vert _{L^2(I)}^2\, ds \right] + C \delta ^{-1} {\mathbb E}\left[ \int \limits _0^t \Vert \partial _x \eta ^\delta \Vert _{L^2(I)}^2\, ds\right] \\&\qquad \le \left( \epsilon ^2 + \frac{\delta }{4} \right) {\mathbb E}\Bigl [\int \limits _0^t \Vert \partial _x \xi ^\delta (s)\Vert _{L^2(I)}^2\, ds \Bigr ] + Ch^2 \delta ^{-2}, \\&{\mathbb E}\left[ \bigl ( \eta ^\delta (t), \xi ^\delta (t) \bigr )_I \right] \!\le \! {\mathbb E}\left[ \frac{1}{4} \Vert \xi ^\delta (t) \Vert _{L^2(I)}^2 \!+\! \Vert \eta ^\delta (t) \Vert _{L^2(I)}^2 \right] \!\le \! \frac{1}{4} {\mathbb E}\left[ \Vert \xi ^\delta (t) \Vert _{L^2(I)}^2\right] \!+ Ch^2, \\&{\mathbb E}\left[ \bigl ( \eta ^\delta (0), \xi ^\delta (t) \bigr )_I \right] \le \frac{1}{4} {\mathbb E}\left[ \Vert \xi ^\delta (t) \Vert _{L^2(I)}^2\right] \!+\! {\mathbb E}\left[ \Vert \eta ^\delta (0) \Vert _{L^2(I)}^2\right] \!\le \!\frac{1}{4} \Vert \xi ^\delta (t) \Vert _{L^2(I)}^2 \!+ Ch^2. \end{aligned}$$

Taking the expectation in (3.12) and using the above estimates then yields

$$\begin{aligned} \sup _{t\in [0,T]} \mathbb {E} \Bigl [\bigl \Vert \xi ^{\delta }(t) \bigr \Vert _{L^2(I)}^2 \Bigr ]&+ 3\delta \, \mathbb {E} \Bigl [\int \limits _0^T \bigl \Vert \partial _x \xi ^\delta (s) \bigr \Vert _{L^2(I)}^2\, ds \Bigr ] \le Ch^2 \bigl ( 1+\delta ^{-2} \bigr ). \end{aligned}$$

(3.13)

Finally, (3.9) follows from the triangle inequality, (3.8), and (3.13). The proof is complete.$\square$

Remark 3.1

• (a) Estimate (3.9) is optimal in the $H^1$-norm, but suboptimal in the $L^2$-norm. The suboptimal rate for the $L^2$-error is caused by the stochastic effect, i.e., the second term on the right-hand side of (3.12), and it is also caused by the lack of the space-time regularity in $L^\infty ((0,T); H^2(I))$ for $u^\delta$.

• (b) The proof still holds if the elliptic projection $R_h^r$ is replaced by the $L^2$-projection $P_h^r$.

We now use estimate (3.13) to derive some stronger norm estimates for $u_h^\delta$. To this end, we define the discrete Laplacian $\partial ^2_h: V_r^h \rightarrow V_r^h$ by

$$\begin{aligned} (\partial ^2_h w_h, v_h)_I = - (\partial _x w_h, \partial _x v_h)_I \qquad \forall w_h,v_h \in V_r^h. \end{aligned}$$

(3.14)

Proposition 3.2

For $\varepsilon \le \sqrt{2(1+\delta )}$ there hold the following estimates for the solution $u_h^{\delta }$ of scheme (3.1):

$$\begin{aligned}&\sup _{0 \le t \le T} {\mathbb E} \Bigl [ \Vert \partial _x u^{\delta }_h(t)\Vert ^2_{L^2(I)}\Bigr ] + \delta \, {\mathbb E} \left[ \int \limits _0^T \Vert \partial ^2_h u_h^{\delta }(s)\Vert ^2_{L^2(I)} \, ds\right] \le C\bigl (1+ \delta ^{-2}\bigr ), \qquad \end{aligned}$$

(3.15)

$$\begin{aligned}&{\mathbb E} \left[ \Vert u^\delta _h(t) - u^\delta _h(s)\Vert _{L^2(I)}^2 + \frac{\delta }{2} \int \limits _s^t \Vert \partial _x[u^\delta _h(\zeta ) - u^\delta _h(s)]\Vert ^2_{L^2(I)} \, d\zeta \right] \\&\quad \le C \bigl (1+ \delta ^{-3}\bigr ) \vert t-s\vert \qquad \forall \, 0 \le s \le t \le T.\nonumber \end{aligned}$$

(3.16)

Proof

Notice that $u_h^\delta =\xi ^\delta +R_h^r u^\delta$ with $\xi ^\delta =u_h^\delta - R_h^r u^\delta \in V_r^h$. By the $H^1$-stability of $R_h^r$, the following inverse inequality for piecewise polynomial function $\xi ^\delta$ (cf. [5]),

$$\begin{aligned} \Vert \partial _x \xi ^\delta (t)\Vert _{L^2(I)} \le Ch^{-1} \Vert \xi ^\delta (t)\Vert _{L^2(I)}, \end{aligned}$$

(2.5), and (3.13) we get

$$\begin{aligned}&\sup _{t \in [0,T]} {\mathbb E} \Bigl [ \Vert \partial _x u_h^{\delta } (t) \Vert ^2_{L^2(I)}\Bigr ] \le 2\sup _{t \in [0,T]} {\mathbb E} \Bigl [ \Vert \partial _x R_h^r u^{\delta } (t) \Vert ^2_{L^2(I)}\Bigr ] \\&\quad + 2 \sup _{t \in [0,T]} {\mathbb E} \Bigl [ \Vert \partial _x \xi ^{\delta } (t) \Vert ^2_{L^2(I)}\Bigr ] \le C\sup _{t \in [0,T]} {\mathbb E} \Bigl [ \Vert \partial _x u^{\delta } (t) \Vert ^2_{L^2(I)}\Bigr ] \\&\quad + \frac{C}{h^2} \sup _{t \in [0,T]} {\mathbb E} \Bigl [ \Vert \xi ^{\delta } (t) \Vert ^2_{L^2(I)}\Bigr ]\le C (1+\delta ^{-2}). \end{aligned}$$

It follows from (3.14) and (3.7) that

$$\begin{aligned} \Vert \partial ^2_h R_h^r w\Vert ^2_{L^2(I)}&= -\bigl ( \partial _x \partial ^2_h R_h^r w , \partial _x R_h^r w\bigr )_{I}\\&= \bigl ( \partial ^2_h R_h^r w, \partial _x^2 w\bigr )_I +\bigl (w-R_h^r w, \partial _h^2R_h^r w\bigr )_I \qquad \forall w \in H^2(I), \end{aligned}$$

and hence by (3.8) we get

$$\begin{aligned} \Vert \partial ^2_h R_h^r w\Vert _{L^2(I)} \le \Vert \partial _x^2 w\Vert _{L^2(I)} +\Vert w-R_h^r w\Vert _{L^2(I)} \le (1+Ch^2) \Vert w\Vert _{H^2(I)}. \end{aligned}$$

(3.17)

By an inverse estimate, (3.17), and (3.13) we have

$$\begin{aligned} {\mathbb E} \left[ \int \limits _0^T \Vert \partial ^2_h u_h^{\delta }(s)\Vert ^2_{L^2(I)}\, ds\right]&\le 2{\mathbb E} \left[ \int \limits _0^T \Bigl (\Vert \partial ^2_h \xi ^{\delta }(s)\Vert ^2_{L^2(I)} + \Vert \partial ^2_h R_h^r u^{\delta }(s)\Vert ^2_{L^2(I)}\Bigr ) ds\right] \\&\le 2{\mathbb E} \left[ \int \limits _0^T \Bigl (Ch^{-2} \Vert \partial _x \xi ^{\delta }(s)\Vert ^2_{L^2(I)} +C\Vert u^{\delta }(s)\Vert ^2_{H^2(I)}\Bigr ) ds\right] \\&\le C \delta ^{-1} (1 + \delta ^{-2}) +C{\mathbb E} \left[ \int \limits _0^T\Vert u^\delta (s) \Vert ^2_{H^2(I)}\, ds\right] , \end{aligned}$$

which together with (2.5) gives the desired bound in (3.15).

To show (3.16), we fix $s\ge 0$ and apply Ito’s formula (cf. [23]) to $f(u_h^\delta ) = \Vert u_h^\delta (t) - u^{\delta }_h(s)\Vert ^2_{L^2(I)}$ to get that

$$\begin{aligned} \Vert u^\delta _h(t) - u^\delta _h(s)&\Vert ^2_{L^2(I)} =- (\epsilon ^2 + 2\delta ) \int \limits _s^t \Bigl ( \partial _x u_h^{\delta }(\zeta ), \partial _x[u^\delta _h(\zeta ) - u^{\delta }_h(s)]\Bigr )_I\, d\zeta \nonumber \\&\qquad \qquad \; - (2-\epsilon ^2) \int \limits _s^t \Bigl ( \mathrm{arctan} \bigl ( \partial _x u_h^{\delta }(\zeta )\bigr ), \partial _x[u^\delta _h(\zeta ) - u^\delta _h(s)]\Bigr )_I\, d\zeta \nonumber \\&\qquad \qquad \; + \epsilon ^2 \int \limits _s^t \Bigl \Vert P_h^r \sqrt{1+ \vert \partial _x u^\delta _h(\zeta ) |^2} \Bigr \Vert ^2_{L^2(I)}\, d\zeta + M_t, \end{aligned}$$

(3.18)

where

$$\begin{aligned} M_t:= \epsilon \int \limits _s^t \left( \sqrt{1+|\partial _x u_h^\delta (\zeta )|^2}, u_h^\delta (\zeta )-u_h^\delta (s) \right) _I dW_\zeta , \end{aligned}$$

which is an $\{ {\mathcal F}_t;\, t \in [s,T]\}$-martingale.

By the $L^2$-stability of $P_h^r$, the triangle and Young’s inequality, and the properties of the square root function, we can bound the third term on the right-hand side as follows:

$$\begin{aligned}&\epsilon ^2\int \limits _s^t\Bigl \Vert P_h^r \sqrt{1+ \bigl |\partial _x \bigl [u^\delta _h(\zeta ) \bigr |^2} \Bigr \Vert ^2_{L^2(I)}\, d\zeta \\&\quad \le \epsilon ^2 \int \limits _s^t \Bigl (\bigl \Vert \partial _x[u^\delta _h(\zeta )- u^\delta _h(s)]\bigr \Vert _{L^2(I)} + \bigl \Vert 1+ \vert \partial _x u^\delta _h(s)\vert \bigr \Vert _{L^2(I)}\Bigr )^2\, d\zeta \nonumber \\&\quad \le \epsilon ^2(1 +\delta ) \int \limits _s^t \Vert \partial _x[u^\delta _h(\zeta ) - u^\delta _h(s)]\bigr \Vert _{L^2(I)}^2\, d\zeta \nonumber \\&\qquad + \epsilon ^2 (4+ \delta ^{-1}) \Bigl ( 1 + \Vert \partial _x u_h^{\delta }(s)\Vert _{L^2(I)}^2\Bigr ) \, |t-s|. \nonumber \end{aligned}$$

Also

$$\begin{aligned}&(\epsilon ^2 + 2\delta ) \int \limits _s^t \Bigl ( \partial _x u^\delta _h(s), \partial _x[u_h^\delta (\zeta ) - u^\partial _h(s)]\Bigr )_I\, d\zeta \\&\quad \le \frac{\delta }{4} \int \limits _s^t \Vert \partial _x[u^\delta _h(\zeta ) - u^\delta _h(s)]\Vert ^2_{L^2(I)}\, d\zeta + (\epsilon ^2+ 2\delta )^2 \delta ^{-1} \, \vert t-s\vert \Vert \partial _x u_h^\delta (s)\Vert ^2_{L^2(I)}, \\&(2-\epsilon ^2) \int \limits _s^t \Bigl ( \mathrm{arctan}\bigl ( \partial _x u_h^\delta (s)\bigr ), \partial _x[u_h^\delta (\zeta ) - u^\partial _h(s)]\Bigr )_I\, d\zeta \\&\quad \le \frac{\delta }{4} \int \limits _s^t \Vert \partial _x[u^\delta _h(\zeta ) - u^\delta _h(s)]\Vert ^2_{L^2(I)}\, d\zeta + 4 (2-\epsilon ^2)^2 \delta ^{-1}\, |t-s|. \end{aligned}$$

Substituting the above estimates into (3.18) yields

$$\begin{aligned}&\Vert u^\delta _h(t)- u_h^\delta (s)\Vert ^2_{L^2(I)} + \frac{\delta }{2} \int \limits _s^t \Vert \partial _x[u^\delta _h(\zeta ) - u^\delta _h(s)]\Vert ^2\, d\zeta \\&\quad \le C \delta ^{-1} \Bigl ((2-\epsilon ^2)^2 + \epsilon ^4 + \delta ^2 \Bigr ) \Bigl ( 1 + \Vert \partial _x u^\delta _h(s)\Vert _{L^2(I)}^2\Bigr ) \, | t-s| + M_t. \end{aligned}$$

Finally, (3.16) follows from applying the expectation to the above inequality and using (3.15) as well as the fact that $\mathbb {E}[M_t]=0$.$\square$

3.2 Fully discrete finite element methods

Let $t_n = n\tau$ for $n=0,1,\ldots ,N$ be a uniform partition of $[0,T]$ with $\tau =T/N$. Our fully discrete finite element method for SPDE (2.3) is defined by seeking an $\{\mathcal {F}_{t_n}; n=0,1,\ldots ,N\}$-adapted $V_r^h$-valued process $\{u_h^n;\, n=0,1,\ldots , N\}$ such that such that $\mathbb {P}$-almost surely

$$\begin{aligned}&\bigl (u_h^{\delta , n+1}, v_h\bigr )_I \!+\!\tau \left( \delta +\frac{\epsilon ^2}{2} \right) \bigl ( \partial _x u_h^{\delta ,n+1}, \partial _x v_h\bigr )_I\nonumber \\&\qquad +\tau \left( 1-\frac{\epsilon ^2}{2}\right) \bigl ( \arctan (\partial _x u_h^{\delta , n+1}), \partial _x v_h \bigr )_I \nonumber \\&\quad = \bigl ( u_h^{\delta ,n}, v_h\bigr )_I + \epsilon \Bigl ( \sqrt{1+|\partial _x u_h^{\delta ,n}|^2}, v_h \Bigr )_I\, \Delta W_{n+1} \quad \forall v_h\in V^h_r, \end{aligned}$$

(3.19)

where $\Delta W_{n+1}:= W(t_{n+1})-W(t_n) \sim {\mathcal N}(0,\tau )$.

We first establish the following stability estimate for $u_h^{\delta ,n}$.

Proposition 3.3

Let $\epsilon \le \sqrt{2(1+\delta )}$. There is a $V_r^h$-valued discrete process $\{u_h^{\delta ,n};\, 0 \le n\le N\}$ which solves scheme (3.19). Moreover, there holds

$$\begin{aligned} \max _{0\le n\le N} \mathbb {E} \Bigl [\bigl \Vert u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ] + 2\delta \sum _{n=0}^{N} \tau \mathbb {E}\Bigl [ \bigl \Vert \partial _x u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ] \le \mathbb {E}\Bigl [\bigl \Vert u_h^{\delta ,0} \bigr \Vert _{L^2(I)}^2\Bigr ] +\epsilon ^2 T. \end{aligned}$$

(3.20)

Proof

The existence of solutions to scheme (3.19) for $\tau ,h > 0$ can be proved by Brouwer’s fixed-point theorem, which uses the coercivity of the operator $I + \tau A_h^\delta$ (see  (3.2)).

To show (3.20), we choose $v_h=u_h^{\delta ,n+1}(\omega )$ in (3.19) to find $\mathbb {P}$-almost surely

$$\begin{aligned}&\frac{1}{2} \Bigl [ \bigl \Vert u_h^{\delta ,n+1} \bigr \Vert _{L^2(I)}^2 -\bigl \Vert u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ] + \frac{1}{2} \bigl \Vert u_h^{\delta ,n+1} - u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \\&\qquad + \tau \left( \delta +\frac{\epsilon ^2}{2}\right) \bigl \Vert \partial _x u_h^{\delta ,n+1} \bigr \Vert _{L^2(I)}^2 + \tau \left( 1-\frac{\epsilon ^2}{2}\right) \left( \arctan (\partial _x u_h^{\delta , n+1}), \partial _x u_h^{\delta , n+1} \right) _I \nonumber \\&\quad =\epsilon \left( \sqrt{1+|\partial _x u_h^{\delta ,n}|^2}, u_h^{\delta ,n} + u_h^{\delta ,n+1}-u_h^{\delta ,n} \right) _I\, \Delta W_{n+1}.\nonumber \end{aligned}$$

(3.21)

We compute

$$\begin{aligned}&\left( \arctan (\partial _x u_h^{\delta , n+1}), \partial _x u_h^{\delta , n+1} \right) _I \ge 0,\\&\quad \epsilon \left( \sqrt{1+|\partial _x u_h^{\delta ,n}|^2}, u_h^{\delta ,n+1}-u_h^{\delta ,n} \right) _I\, \Delta W_{n+1} \\&\quad \le \frac{1}{2} \bigl \Vert u_h^{\delta ,n+1} - u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 + \frac{\epsilon ^2}{2} \Bigl \Vert \sqrt{1+ |\partial _x u_h^{\delta ,n}|^2} \Bigr \Vert _{L^2(I)}^2 |\Delta W_{n+1}|^2. \end{aligned}$$

The last estimate controls one part of the stochastic term in (3.21), while the expectation of the remaining part vanishes. By the tower property for expectations, there holds

$$\begin{aligned} \frac{\epsilon ^2}{2} {\mathbb E} \left[ \Bigl \Vert \sqrt{1+ \vert \partial _x u_h^{\delta ,n}\vert ^2} \Bigr \Vert _{L^2(I)}^2 {\mathbb E}\bigl [\vert \Delta W_{n+1}\vert ^2\vert {\mathcal F}_{t_n}\bigr ] \right] = \frac{\epsilon ^2}{2} \tau \, {\mathbb E} \left[ 1 + \Vert \partial _x u^{\delta ,n}\Vert ^2_{L^2(I)}\right] , \end{aligned}$$

such that we get

$$\begin{aligned}&\frac{1}{2} \mathbb {E}\, \Bigl [ \bigl \Vert u_h^{\delta ,n+1} \bigr \Vert _{L^2(I)}^2 -\bigl \Vert u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ] + \tau \delta \, \mathbb {E} \Bigl [\bigl \Vert \partial _x u_h^{\delta ,n+1} \bigr \Vert _{L^2(I)}^2 \Bigr ] \\&\qquad +\, \frac{\epsilon ^2}{2} \tau \, {\mathbb E} \Bigl [ \bigl \Vert \partial _x u_h^{\delta ,n+1} \bigr \Vert _{L^2(I)}^2 -\bigl \Vert \partial _x u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2\Bigr ] \le \epsilon ^2 \tau .\nonumber \end{aligned}$$

(3.22)

After summation, we arrive at

$$\begin{aligned} \max _{0\le n\le N} \mathbb {E} \left[ \bigl \Vert u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \right] + 2\delta \tau \sum _{n=0}^{N} \mathbb {E} \Bigl [ \bigl \Vert \partial _x u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ] \le \mathbb {E}\Bigl [\bigl \Vert u_h^{\delta ,0} \bigr \Vert _{L^2(I)}^2\Bigr ] +\epsilon ^2 T. \end{aligned}$$

So (3.20) holds. The proof is complete.$\square$

Next, we derive an error bound for $u_h^\delta (t_n)-u_h^{\delta ,n}$.

Theorem 3.2

Let $r=1$. There holds the following error estimate:

$$\begin{aligned}&\sup _{0\le n\le N} \mathbb {E} \Bigl [\bigl \Vert u_h^\delta (t_n) - u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ] + \delta \, \mathbb {E} \Bigl [\sum _{n=0}^N \tau \bigl \Vert \partial _x u_h^\delta (t_n) - \partial _x u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ] \\&\quad \le CT\bigl (1+ \delta ^{-2} \bigr ) h^{-2} \tau . \nonumber \end{aligned}$$

(3.23)

Proof

Let $e^{\delta ,n}:=u_h^\delta (t_n)-u_h^{\delta ,n}$. It follows from (3.1) that for all $\{ t_n; n \ge 0\}$ there holds ${\mathbb P}$-almost surely

$$\begin{aligned}&\left( u_h^\delta (t_{n+1}), v_h \right) _I - \bigl ( u_h^\delta (t_n), v_h \bigr )_I \\&\quad = -\left( \delta +\frac{\epsilon ^2}{2}\right) \int \limits _{t_n}^{t_{n+1}} \bigl ( \partial _x u_h^\delta (s), \partial _x v_h \bigr )_I ds \nonumber \\&\qquad \, -\left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _{t_n}^{t_{n+1}} \bigl ( \arctan (\partial _x u_h^\delta (s)), \partial _x v_h \bigr )_I ds \nonumber \\&\qquad \, + \epsilon \int \limits _{t_n}^{t_{n+1}} \left( \sqrt{1+|\partial _x u_h^\delta (s)|^2}, v_h\right) _I dW_s \quad \forall v_h\in V_r^h . \nonumber \end{aligned}$$

(3.24)

Subtracting (3.19) from (3.24) yields the following error equation:

$$\begin{aligned}&\left( e^{\delta ,n+1}, v_h \right) _I - \left( e^{\delta , n}, v_h \right) _I \nonumber \\&\quad = -\bigl (\delta +\frac{\epsilon ^2}{2}\bigr ) \int \limits _{t_n}^{t_{n+1}} \bigl ( \partial _x u_h^\delta (s)-\partial _x u_h^{\delta ,n+1}, \partial _x v_h \bigr )_I ds \nonumber \\&\qquad -\bigl ( 1-\frac{\epsilon ^2}{2}\bigr ) \int \limits _{t_n}^{t_{n+1}} \Bigl ( \arctan \bigl (\partial _x u_h^\delta (s) \bigr )-\arctan (\partial _x u_h^{\delta ,n+1}), \partial _x v_h \Bigr )_I ds \nonumber \\&\qquad + \epsilon \int \limits _{t_n}^{t_{n+1}} \Bigl ( \sqrt{1+|\partial _x u_h^\delta (s)|^2} - \sqrt{1+|\partial _x u_h^{\delta , n}|^2}, v_h\Bigr )_I dW_s . \end{aligned}$$

(3.25)

Choosing $v_h=e^{\delta ,n+1}(\omega )$ in (3.25) leads to $\mathbb {P}$-almost surely

$$\begin{aligned}&\frac{1}{2}\left[ \Vert e^{\delta ,n+1} \Vert _{L^2(I)}^2 - \Vert e^{\delta ,n} \Vert _{L^2(I)}^2 \right] +\frac{1}{2} \left\| e^{\delta ,n+1} - e^{\delta ,n} \right\| _{L^2(I)}^2 \nonumber \\&\qquad + \left( \frac{\epsilon ^2}{2} + \delta \right) \tau \, \Vert \partial _x e^{\delta ,n+1}\Vert ^2_{L^2(I)} \nonumber \\&\quad = - \left( \frac{\epsilon ^2}{2}+\delta \right) \int \limits _{t_n}^{t_{n+1}} \Bigl ( \partial _x u_h^\delta (s)- \partial _x u_h^\delta (t_{n+1}) , \partial _x e^{\delta ,n+1} \Bigr )_I ds \nonumber \\&\qquad -\left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _{t_n}^{t_{n+1}} \Bigl ( \arctan \bigl (\partial _x u_h^\delta (s) \bigr ) -\mathrm{arctan} \bigl ( \partial _x u_h^{\delta }(t_{n+1})\bigr ) \nonumber \\&\qquad + \mathrm{arctan} \bigl ( \partial _x u_h^{\delta }(t_{n+1})\bigr ) -\arctan (\partial _x u_h^{\delta ,n+1}), \partial _x e^{\delta , n+1} \Bigr )_I ds \nonumber \\&\qquad + \epsilon \int \limits _{t_n}^{t_{n+1}} \Bigl ( \sqrt{1+|\partial _x u_h^\delta (s)|^2} - \sqrt{1+|\partial _x u_h^{\delta , n}|^2}, e^{\delta , n+1} \Bigr )_I dW_s. \end{aligned}$$

(3.26)

We now bound each term on the right-hand side. First, since ${\mathbb E}[\Delta W_{n+1}] = 0$, by Ito’s isometry, the inequality $\bigl (\sqrt{1+x^2} - \sqrt{1+y^2} \bigr )^2 \le (x-y)^2$, and the inverse inequality we get

$$\begin{aligned}&\mathbb {E}\left[ \epsilon \int \limits _{t_n}^{t_{n+1}} \left( \sqrt{1+|\partial _x u_h^\delta (s)|^2} - \sqrt{1+|\partial _x u_h^{\delta , n}|^2}, e^{\delta , n+1} \right) _I dW_s \right] \nonumber \\&\quad \le {\mathbb E} \left[ \frac{1}{2}\Vert e^{\delta ,n+1} - e^{\delta ,n}\Vert ^2_{L^2(I)}\right] + \frac{\epsilon ^2}{2} \int \limits _{t_n}^{t_{n+1}} \Vert \partial _x\left[ u_h^{\delta }(s) - u_h^{\delta ,n} \right] \Vert ^2_{L^2(I)} \, ds\Bigr ] \nonumber \\&\quad \le \frac{1}{2} {\mathbb E} \left[ \Vert e^{\delta ,n+1} - e^{\delta ,n}\Vert ^2_{L^2(I)}\right] +{\mathbb E} \left[ \left( \frac{\epsilon ^2}{2} + \frac{\delta }{2} \right) \tau \, \Vert \partial _x e^{\delta ,n}\Vert ^2_{L^2(I)}\right. \nonumber \\&\qquad \left. + \left( \frac{\epsilon ^2}{2} + \frac{2}{\delta } \right) \int \limits _{t_n}^{t_{n+1}} \Vert \partial _x[u_h^{\delta }(s) - u_h^\delta (t_n)]\Vert ^2_{L^2(I)} \, ds\right] \nonumber \\&\quad \le \frac{1}{2} {\mathbb E} \Bigl [ \Vert e^{\delta ,n+1} - e^{\delta ,n}\Vert ^2_{L^2(I)}\Bigr ] +\left( \frac{\epsilon ^2}{2} + \frac{\delta }{2} \right) \tau \, {\mathbb E} \Bigl [ \Vert \partial _x e^{\delta ,n}\Vert ^2_{L^2(I)}\Bigr ]\nonumber \\&\qquad + C\bigl ( 1+\delta ^{-1} \bigr ) h^{-2} \, \mathbb {E} \left[ \int \limits _{t_n}^{t_{n+1}} \Vert u_h^{\delta }(s) - u_h^\delta (t_n)\Vert ^2_{L^2(I)} \, ds\right] \!. \end{aligned}$$

(3.27)

An elementary calculation and an application of the inverse inequality yield

$$\begin{aligned}&\left( \frac{\epsilon ^2}{2}+\delta \right) \int \limits _{t_n}^{t_{n+1}} \Bigl ( \partial _x u_h^\delta (s)- \partial _x u_h^\delta (t_{n+1}) , \partial _x e^{\delta ,n+1} \Bigr )_I ds \nonumber \\&\quad \le \frac{\delta }{8} \tau \, \Vert \partial _x e^{\delta ,n+1}\Vert ^2_{L^2(I)} + \frac{2 (\frac{\epsilon ^2}{2} + \delta )^2}{\delta } \int \limits _{t_n}^{t_{n+1}} \Vert \partial _x[u^\delta _h(s) - u^\delta _h(t_{n+1})]\Vert ^2_{L^2(I)}\, ds\nonumber \\&\quad \le \frac{\delta }{8} \tau \, \Vert \partial _x e^{\delta ,n+1}\Vert ^2_{L^2(I)} +(\epsilon ^2+2\delta )^2 \delta ^{-1} h^{-2} \int \limits _{t_n}^{t_{n+1}} \Vert u_h^{\delta }(s) - u_h^\delta (t_n)\Vert ^2_{L^2(I)} \, ds. \end{aligned}$$

(3.28)

By the monotonicity of $\arctan$ we get

$$\begin{aligned}&-\left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _{t_n}^{t_{n+1}} \Bigl ( \arctan \bigl (\partial _x u_h^\delta (s) \bigr ) - \mathrm{arctan} \bigl ( \partial _x u_h^{\delta }(t_{n+1})\bigr ) \nonumber \\&\quad +\mathrm{arctan} \bigl ( \partial _x u_h^{\delta }(t_{n+1})\bigr ) -\arctan (\partial _x u_h^{\delta ,n+1}), \partial _x e^{\delta , n+1} \Bigr )_I ds \nonumber \\&\le -\left( 1-\frac{\epsilon ^2}{2}\right) \int \limits _{t_n}^{t_{n+1}} \Bigl ( \arctan \bigl (\partial _x u_h^\delta (s) \bigr ) - \mathrm{arctan} \bigl ( \partial _x u_h^{\delta }(t_{n+1})\bigr ), \partial _x e^{\delta ,n+1} \Bigr ) \, ds \nonumber \\&\le \frac{\delta }{8} \tau \, \Vert \partial _x e^{\delta ,n+1}\Vert ^2_{L^2(I)} + \frac{2}{\delta }(1-\frac{\epsilon ^2}{2})^2 \int \limits _{t_n}^{t_{n+1}} \Vert \partial _x[u^\delta _h(s) - u^\delta _h(t_{n+1})]\Vert ^2_{L^2(I)}\, ds \nonumber \\&\le \frac{\delta }{8} \tau \, \Vert \partial _x e^{\delta ,n+1}\Vert ^2_{L^2(I)} + (2-\epsilon ^2)^2 \delta ^{-1} h^{-2} \int \limits _{t_n}^{t_{n+1}} \Vert u_h^{\delta }(s) - u_h^\delta (t_n)\Vert ^2_{L^2(I)} \, ds. \end{aligned}$$

(3.29)

Finally, substituting the above estimates into (3.26), summing over $n=0,1, 2, \ldots , N-1$, and using (3.16) and the fact that $e^{\delta ,0} = 0$ we get

$$\begin{aligned}&\sup _{0 \le n \le N} {\mathbb E}\left[ \Vert e^{\delta ,n}\Vert ^2_{L^2(I)}\right] + \frac{\delta }{2} {\mathbb E} \left[ \tau \sum _{n=0}^N \Vert \partial _x e^{\delta ,n+1}\Vert ^2_{L^2(I)} \right] \\&\quad \le C\bigl ( 1+\delta ^{-1}\bigr ) h^{-2} \sum _{n=0}^N \int \limits _{t_n}^{t_{n+1}} \sup \limits _{s\in [t_n, t_{n+1}]} \mathbb {E}\left[ \Vert u_h^{\delta }(s) - u_h^\delta (t_n)\Vert ^2_{L^2(I)} \right] \, ds\\&\quad \le CT(1+\delta ^{-1})^2 h^{-2} \tau , \end{aligned}$$

which infers (3.23). The proof is complete.$\square$

We conclude this section by stating the following error estimates for the fully discrete finite element solution $u_h^{\delta ,n}$ as an approximation to the solution of the original mean curvature flow equation (1.5).

Theorem 3.3

Let $u$ and $u_h^{\delta , n}$ denote respectively the solutions of SPDE (1.5) and scheme (3.19). Under assumptions of Theorems 2.1, 3.1, and 3.2, there holds the following error estimate:

$$\begin{aligned}&\sup _{0\le n\le N} \mathbb {E} \Bigl [\bigl \Vert u(t_n) - u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ] +\delta \, \mathbb {E} \left[ \sum _{n=0}^N \tau \bigl \Vert \partial _x u(t_n) -\partial _x u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \right] \\&\quad \le CT\delta + C\bigl (1+\delta ^{-2} \bigr ) h^2 + CT\bigl (1+ \delta ^{-2} \bigr ) h^{-2} \tau . \nonumber \end{aligned}$$

(3.30)

Inequality (3.30) follows immediately from Theorems 2.3, 3.1 and 3.2, and an application of the triangle inequality.

Remark 3.2

Again, we note that the main reason to have a restrictive coupling between numerical parameters in (3.30) is due to the lack of Hölder continuity (in time) estimate for $\partial _x u_h^\delta$ in $L^2$-norm. On the other hand, it can be shown that, under a stronger regularity assumption, the estimate (3.30) can be improved to

$$\begin{aligned}&\sup _{0\le n\le N} \mathbb {E} \Bigl [\bigl \Vert u(t_n) - u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \Bigr ]\\&\quad + \delta \, \mathbb {E} \left[ \sum _{n=0}^N \tau \bigl \Vert \partial _x u(t_n) -\partial _x u_h^{\delta ,n} \bigr \Vert _{L^2(I)}^2 \right] \le C\bigl ( h^2 + \tau +\delta \bigr )\, . \nonumber \end{aligned}$$

(3.31)

This is because we no longer need to use the inverse inequality to get (3.27)–(3.29), and (3.31) can be obtained by starting with a control of the time discretization first.

4 Numerical experiments

In this section we shall first present some numerical experiments to gauge the performance of the proposed fully discrete finite element method and to examine the effect of the noise for long-time dynamics of the stochastic MCF of planar graphs, and we then present a numerical study of the stochastic MCF driven by both colored and space-time white noises where no theoretical result is known so far in the literature. We like to note that all our numerical experiments are done in Matlab. At each time step, a nonlinear algebraic system must be solved, which is done by using Matlab’s built-in Newton solver in all our numerical tests. In addition, all space norms are computed approximately using sufficiently high order numerical quadrature formulas.

Table 1 Computed time discretization errors and convergence rates

4.1 Verifying the rate of convergence of time discretization

To verify the rate of convergence of the time discretization obtained in Theorem 3.3, in this first test we use the following parameters $\epsilon =1$, $\delta =10^{-5}$, and $T=0.1$. In order to computationally generate a driving reference ${\mathbb R}$-valued Wiener process, we use the smaller time step $\tau =10^{-5}$. The initial condition is set to be $u_0(x)=\sin (\pi x)$. To calculate the rate, we compute the solution $u_h^{\delta ,n}$ for varying $\tau =0.0005, 0.001, 0.002, 0.004$. We take $500$ stochastic samples at each time step $t_n$ in order to compute the expected values of the $L^{\infty }((0,T);L^2(I))$-norm of the error. The computed errors along with the computed convergence rates are exhibited in Table 1 and Fig. 1. The numerical results confirm the theoretical result of Theorem 3.2. In Fig. 2, we plot the errors of the computed solution with regularization (i.e., $\delta >0$) and without regularization (i.e., $\delta =0$). The comparison shows that without the regularization term our numerical methods still compute correct solutions for some problems although our convergence theory requires that $\delta >0$.

Fig. 1

Plot of the errors in Table 1

Fig. 2

Comparison of the computed solution with (blue line) and without (red line) the regularization term (Color figure online)

4.2 Dynamics of the stochastic MCF

We shall perform several numerical tests to demonstrate the dynamics of the stochastic MCF with different magnitudes of noise (i.e., different sizes of the parameter $\epsilon$).

Figure 3 shows the surface plots of the computed solution $u_h^{\delta ,n}$ at one stochastic sample over the space-time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right) with the initial value $u_0(x)=\sin (\pi x)$ and the noise intensity parameter $\epsilon =0.1$. The test shows that the solution converges to a steady state solution at the end.

Fig. 3

Surface plots of computed solution at a fixed stochastic sample on the space time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right). $u_0(x)=\sin (\pi x)$ and $\epsilon =0.1$

Figures 4–6 are the counterparts of Fig. 3 with noise intensity parameter $\epsilon =1, \sqrt{2}, 5$, respectively. We note that the error estimate of Theorem 3.3 does not apply to the latter case because the condition $\epsilon \le \sqrt{2(1+ \delta )}$ is violated. However, the computation result suggests that the stochastic MCF also converges to the steady state solution at the end although the paths to reach the steady state are different for different noise intensity parameter $\epsilon$.

Fig. 4

Surface plots of computed solution at a fixed stochastic sample on the space time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right). $u_0(x)=\sin (\pi x)$ and $\epsilon =1$

Fig. 5

Surface plots of computed solution at a fixed stochastic sample on the space time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right). $u_0(x)=\sin (\pi x)$ and $\epsilon =\sqrt{2}$

Fig. 6

Surface plots of computed solution at a fixed stochastic sample on the space time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right). $u_0(x)=\sin (\pi x)$ and $\epsilon =5$

We then repeat the above four tests after replacing the smooth initial function $u_0$ by the following non-smooth initial function:

$$\begin{aligned} u_0(x)={\left\{ \begin{array}{ll} 10x, &{} \text {if}\ x\le 0.25, \\ 5-10x, &{} \text {if}\ 0.25<x\le 0.5, \\ 10x-5, &{} \text {if}\ 0.5<x\le 0.75, \\ 10-10x, &{}\text {if}\ 0.75<x\le 1. \\ \end{array}\right. } \end{aligned}$$

(4.1)

The surface plots of the computed solutions are shown in Figs. 7–10, respectively. Again, the numerical results suggest that the solution of the stochastic MCF converges to the steady state solution at the end although the paths to reach the steady state are different for different noise intensity parameter $\epsilon$. As expected, the geometric evolution dominates for small $\epsilon$, but the noise dominates the geometric evolution for large $\epsilon$.

Fig. 7

Surface plots of computed solution at a fixed stochastic sample on the space time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right). $u_0$ is given in (4.1) and $\epsilon =0.1$

Fig. 8

Surface plots of computed solution at a fixed stochastic sample on the space time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right). $u_0$ is given in (4.1) and $\epsilon =1$

Fig. 9

Surface plots of computed solution at a fixed stochastic sample on the space time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right). $u_0$ is given in (4.1) and $\epsilon =\sqrt{2}$

Fig. 10

Surface plots of computed solution at a fixed stochastic sample on the space time domains $(0,1)\times (0,0.1)$ (left) and $(0,1)\times (0, 2^8\times 10^{-5})$ (right). $u_0$ is given in (4.1) and $\epsilon =5$

4.3 Verifying energy dissipation

It follows from (2.5) that the “energy” $J(t):=\frac{1}{2} \mathbb {E}\left[ \Vert \partial _x u^{\delta }(t)\Vert _{L^2(I)}^2\right]$ decreases monotonically in time. In the following we verify this fact numerically. Again, we consider the case with the initial function $u_0(x)=\sin (\pi x)$ and the noise intensity parameter $\epsilon =1$. It is not hard to prove that $J(t)$ converges to zero as $t\rightarrow \infty$. Figure 11 plots the computed $J(t)$ as a function of $t$. The numerical result suggests that $J(t)$ does not change anymore for $t\ge 0.1$.

4.4 Thresholding for colored noise

In this subsection we present a computational study of the interplay of noise and geometric evolution in (1.5), which is beyond our theoretical results in Sects. 3.1 and 3.2. For this purpose, we use driving colored noise represented by the $Q$-Wiener process ($J \in {\mathbb N}$)

$$\begin{aligned} W_t = \sum _{j=1}^{J} q_j^{\frac{1}{2}} \beta _j(t) e_j\,, \end{aligned}$$

(4.2)

where $\{ \beta _j(t);\, t \ge 0\}_{j\ge 1}$ denotes a family of real-valued independent Wiener processes on $\bigl ( \Omega , {\mathcal F}, {\mathbb F}, {\mathbb P}\bigr )$, and $\{ (q_j, e_j)\}_{j=1}^{J}$ is an eigen-system of the symmetric, non-negative trace-class operator $Q: L^2(I) \rightarrow L^2(I)$, with $e_j = \sqrt{2} \sin (j \pi x)$. In particular, we like to numerically address the following questions:

• (A) Thresholding: By Theorem 2.1, strong solutions of (1.5) exist for $\varepsilon \le \sqrt{2}$, and a similar result can be shown for the PDE problem with the noise (4.2). What are admissible intensities of the noise suggested by computations? Moreover, what do the computations suggest about the stochastic MCF in the case of spatially white noise (i.e., $q_j\equiv 1, J=\infty$) where no theoretical result is available so far?

• (B) General initial profiles: The deterministic evolution of Lipschitz initial graphs is well-understood. For example, the (upper) graph of two touching spheres may trigger non-uniqueness. What are the regularization and the noise excitation effects in the case of the initial data with infinite energy and using different noises?

Recall that the estimate in Proposition 3.3 for $V_r^h$-valued solution $u^{\delta ,n}_h$ suggests that $\varepsilon >0$ ought be sufficiently small to ensure the existence. In our test, we employ the colored noise (4.2) with $q_j^{\frac{1}{2}} = j^{-0.6}$, $J = 20$, and the following non-Lipschitz initial data:

$$\begin{aligned} u_0(x) = |0.5 - x |^\kappa \qquad \forall \, x \in (0,1)\,, \end{aligned}$$

(4.3)

where $\kappa = 0.1$. In addition, we set $(\tau , h) = (0.01, 0.02)$ and $T = \frac{1}{2}$. Figure 12 shows the single trajectory of the stochastic MCF plotted as graphs over the space-time domain with, respectively, $\epsilon =0.1, 0.5, \sqrt{2}$. The results indicate thresholding, namely, the trajectories grow rapidly in time for sufficiently large values $\varepsilon$, and the noise effect dominates the geometric evolution. The excitation effect of the noise on the geometric evolution is illustrated by corresponding plots for the evolution of the functional $n \mapsto \Vert \partial _x u_h^{\delta ,n}(\omega )\Vert ^2_{L^2}$ vs its expectation $n \mapsto {\mathbb E}\bigl [ \Vert \partial _x u_h^{\delta ,n}\Vert ^2_{L^2}\bigr ]$ in Figs. 13 and 14. We observe that the geometric evolution dominates for small values of $\varepsilon$, while the noise evolution takes over for large values of $\varepsilon$.

Fig. 11

Decay of the energy $J(t)$ on the interval $(0, 0.1)$

Fig. 12

Thresholding for colored noise: Trajectories for $\varepsilon = 0.1$ (top left), $\varepsilon = 0.5$ (top right), $\varepsilon = \sqrt{2}$ (bottom)

Fig. 13

Geometric evolution vs colored noise evolution ($q^{\frac{1}{2}}_j = j^{-0.6}$, $J=20$): 1st row: single trajectory for $n \mapsto \Vert \partial _x u_h^{\delta ,n}(\omega ) \Vert ^2_{L^2}$ and $\varepsilon = 0.1$ (left), $\varepsilon = 0.5$ (right); 2nd row: $n \mapsto {\mathbb E}\bigl [ \Vert \partial _x u_h^{\delta ,n}\Vert ^2_{L^2}\bigr ]$ for $\varepsilon = 0.1$ (left), $\varepsilon = 0.5$ (right)

Fig. 14

Geometric evolution vs colored noise evolution ($q^{\frac{1}{2}}_j = j^{-1}$, $J=50$): 1st row: single trajectory for $n \mapsto \Vert \partial _x u_h^{\delta ,n}(\omega ) \Vert ^2_{L^2}$ and $\varepsilon = 0.1$ (left), $\varepsilon = 0.5$ (right); 2nd row: $n \mapsto {\mathbb E}\bigl [ \Vert \partial _x u_h^{\delta ,n}\Vert ^2_{L^2}\bigr ]$ for $\varepsilon = 0.1$ (left), $\varepsilon = 0.5$ (right)

4.5 Thresholding for white noise

We now consider the case of white noise in (3.19), that is, $q_j \equiv 1$ in (4.2) and $J = \infty$, for which the solvability of (1.3) is not known. Figure 15 shows the single trajectory of the stochastic MCF (with the same data as in Sect. 4.4) plotted as graphs over the space-time domain with, respectively, $\epsilon =0.1, 0.5, \sqrt{2}$. We observe a very rapid growth of trajectories (numerical values range between $10^{14}$ and $10^{21}$) even for small values of $\varepsilon >0$. These numerical results suggest either a rapid growth or a finite time explosion for the stochastic MCF in the case of white noise.

Fig. 15

Thresholding and white noise: $\varepsilon = 0.1$ (top left), $\varepsilon = 0.5$ (top right) $\varepsilon = \sqrt{2}$ (bottom)