Correction to: Convergent numerical approximation of the stochastic total variation flow

1 Correction to: Stoch PDE: Anal Comp (2021) 9:437–471 https://doi.org/10.1007/s40072-020-00169-4

2 Introduction

Let ${\mathcal {O}}\subset {\mathbb {R}}^d$ be an open convex domain with piecewise smooth boundary. We consider numerical approximation of the stochastic total variation flow

$$\begin{aligned} \,\mathrm {d}X&= \mathrm {div}\left( \frac{\nabla X}{\left| \nabla X\right| }\right) \,\mathrm {d}t -\lambda (X - g) \,\mathrm {d}t +X\,\mathrm {d}W,&{\text {in }} (0,T)\times {\mathcal {O}}, \nonumber \\ X&= 0&{\text {on }} (0,T)\times \partial {\mathcal {O}}, \nonumber \\ X(0)&=x_0&{\text {in }} {\mathcal {O}}, \end{aligned}$$

(1)

which is constructed via the discretization of the regularized problem

$$\begin{aligned} \,\mathrm {d}X^{\varepsilon }&= \mathrm {div}\left( \frac{\nabla X^{\varepsilon }}{\sqrt{|\nabla X^{\varepsilon }|^2+\varepsilon ^2}}\right) \,\mathrm {d}t-\lambda (X^{\varepsilon }-g)\,\mathrm {d}t+X^{\varepsilon }\,\mathrm {d}W&{\text {in }} (0,T)\times {\mathcal {O}}, \nonumber \\ X^{\varepsilon }&= 0&{\text {on }} (0,T)\times \partial {\mathcal {O}}, \nonumber \\ X^{\varepsilon }(0)&=x_0&{\text {in }} {\mathcal {O}}. \end{aligned}$$

(2)

Throughout the paper we employ the notation from [4]. The first error is corrected in Sect. 2 and the correction of the second error is provided in Sect. 3.

3 Definition of the SVI solution and the uniqueness proof

In the proof of [4,  Theorem 3.1] the term IV in (29) is wrongly rewritten as

$$\begin{aligned} IV = -\left( X_1^{\varepsilon }-X_{2,n}^{\varepsilon ,\delta },\mathrm {div}\frac{\nabla X_{2,n}^{\varepsilon ,\delta }}{ \sqrt{ \vert \nabla X_{2,n}^{\varepsilon ,\delta } \vert ^2 +\varepsilon ^2}} \right) =\left( \nabla X_1^{\varepsilon }-\nabla X_{2,n}^{\varepsilon ,\delta }, \frac{\nabla X_{2,n}^{\varepsilon ,\delta }}{ \sqrt{ \vert \nabla X_{2,n}^{\varepsilon ,\delta } \vert ^2 +\varepsilon ^2}} \right) \, \end{aligned}$$

since $X_1^{\varepsilon }$ is only in $BV({\mathcal {O}})$ and possibly non-zero at the boundary. Hence, to show the uniqueness of the SVI solutions for $\varepsilon >0$ requires a modification of the definition which takes into account the value at the solution at the boundary. This definition is consistent with the one from [2], which also shows uniqueness in case $\varepsilon =0$.

We define the following functionals which include the corresponding boundary terms

$$\begin{aligned} \bar{ {\mathcal {J}}}_{\varepsilon ,\lambda }(u)= {\left\{ \begin{array}{ll} {\mathcal {J}}_{\varepsilon ,\lambda }(u) + \int _{\partial {\mathcal {O}}} \left| \gamma _0(u)\right| \,\mathrm {d}{\mathcal {H}}^{n-1} ~~ &{} {\text {for}}~ u \in BV({\mathcal {O}})\cap L^2({\mathcal {O}}),\\ +\infty ~~ &{}{\text {for}}~ u \in BV({\mathcal {O}}){\setminus } L^2({\mathcal {O}}), \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \bar{ {\mathcal {J}}}_\lambda (u)= {\left\{ \begin{array}{ll} {\mathcal {J}}_\lambda (u) + \int _{\partial {\mathcal {O}}} \left| \gamma _0(u)\right| \,\mathrm {d}{\mathcal {H}}^{n-1} ~~ &{} {\text {for}}~ u \in BV({\mathcal {O}})\cap L^2({\mathcal {O}}),\\ +\infty ~~ &{} {\text {for}}~ u \in BV({\mathcal {O}}){\setminus } L^2({\mathcal {O}}), \end{array}\right. } \end{aligned}$$

where $\gamma _0(u)$ is the trace of $u\in BV({\mathcal {O}})$ on the boundary and $\,\mathrm {d}{\mathcal {H}}^{n-1}$ is the Hausdorff measure on $\partial {\mathcal {O}}$. $\bar{{\mathcal {J}}}_{\varepsilon ,\lambda }$ and $\bar{{\mathcal {J}}}_{\lambda }$ are both convex and lower semicontinuous on $L^2({\mathcal {O}})$ and the lower semicontinuous hulls of $\bar{{\mathcal {J}}}_{\varepsilon ,\lambda }\vert _{{\mathbb {H}}^1_0}$ or $\bar{{\mathcal {J}}}_{\lambda }\vert _{{\mathbb {H}}^1_0}$ respectively, cf. [1,  Proposition 11.3.2]. We define the SVI solution as follows.

Definition 2.1

Let $0< T < \infty$, $\varepsilon \in [0,1]$ and $x_0 \in L^2(\Omega ,{\mathcal {F}}_0;{\mathbb {L}}^2)$ and $g \in {\mathbb {L}}^2$. Then a $({\mathcal {F}}_t)$-adapted map $X^{\varepsilon }\in L^2(\Omega ; C([0,T];{\mathbb {L}}^2))\cap L^1(\Omega ; L^1((0,T);BV({\mathcal {O}})))$ (denoted by $X \in L^2(\Omega ; C([0,T];{\mathbb {L}}^2))\cap L^1(\Omega ; L^1((0,T);BV({\mathcal {O}})))$ for $\varepsilon =0$) is called an SVI solution of (2) (or (1) if $\varepsilon =0$) if $X^{\varepsilon }(0)=x_0$ ($X(0)=x_0$), and for each $({\mathcal {F}}_t)$-progressively measurable process $G\in L^2(\Omega \times (0,T),{\mathbb {L}}^2)$ and for each $({\mathcal {F}}_t)$-adapted ${\mathbb {L}}^2$-valued process Z with ${\mathbb {P}}$-a.s. continuous sample paths, s.t. $Z \in L^2(\Omega \times (0,T);{\mathbb {H}}^1_0)$, which together satisfy the equation

$$\begin{aligned} \,\mathrm {d}Z(t)= -G(t) \,\mathrm {d}t +Z(t)\,\mathrm {d}W(t), ~ t\in [0,T], \end{aligned}$$

it holds for $\varepsilon \in (0,1]$ that

$$\begin{aligned} \frac{1}{2}&{\mathbb {E}}\left[ \Vert X^{\varepsilon }(t)-Z(t)\Vert ^2\right] +{\mathbb {E}}\left[ \int _0^t{\bar{{\mathcal {J}}}_{\varepsilon ,\lambda }}(X^{\varepsilon }(s)) \,\mathrm {d}s\right] \nonumber \\&\le \frac{1}{2} {\mathbb {E}}\left[ \Vert x_0-Z(0)\Vert ^2\right] +{\mathbb {E}}\left[ \int _0^t{ \bar{{\mathcal {J}}}_{\varepsilon ,\lambda }}(Z(s)) \,\mathrm {d}s\right] \nonumber \\&+ \frac{1}{2}{\mathbb {E}}\left[ \int _0^t\Vert X^{\varepsilon }(s)-Z(s)\Vert ^2 \,\mathrm {d}s\right] +{\mathbb {E}}\left[ \int _0^t\left( X^{\varepsilon }(s)-Z(s),G \right) \,\mathrm {d}s\right] , \end{aligned}$$

(3)

and analogously for $\varepsilon =0$ it holds that

$$\begin{aligned} \frac{1}{2}&{\mathbb {E}}\left[ \Vert X(t)-Z(t)\Vert ^2\right] +{\mathbb {E}}\left[ \int _0^t{ \bar{{\mathcal {J}}}_{\lambda }}(X(s)) \,\mathrm {d}s\right] \nonumber \\&\le \frac{1}{2} {\mathbb {E}}\left[ \Vert x_0-Z(0)\Vert ^2\right] +{\mathbb {E}}\left[ \int _0^t{ \bar{{\mathcal {J}}}_{\lambda }}(Z(s)) \,\mathrm {d}s\right] \nonumber \\&+\frac{1}{2} {\mathbb {E}}\left[ \int _0^t\Vert X(s)-Z(s)\Vert ^2 \,\mathrm {d}s\right] +{\mathbb {E}}\left[ \int _0^t\left( X(s)-Z(s),G \right) \,\mathrm {d}s\right] . \end{aligned}$$

(4)

The existence of SVI solutions (3), (4) follows as in [4,  Theorem 3.1] by the lower semicontinuity of $\bar{ {\mathcal {J}}}_{\varepsilon , \lambda }$, $\bar{ {\mathcal {J}}}_\lambda$, respectively. The uniqueness of SVI solution (4) follows from [2,  Theorem 3.2].

To show uniqueness of the SVI solution (3) we proceed as in [4,  Theorem 3.1] with exception that the term IV in (29) takes a different form. In particular, to obtain uniqueness we have to show the following estimate:

$$\begin{aligned} IV:=\left( X_1^{\varepsilon }-X^{\varepsilon ,\delta }_{2,n},-\mathrm {div}\frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}} \right) \le \bar{{\mathcal {J}}}_{\varepsilon ,0}(X^{\varepsilon }_1)-\bar{{\mathcal {J}}}_{\varepsilon ,0}(X^{\varepsilon ,\delta }_{2,n}). \end{aligned}$$

(5)

We note that term IV is well defined since $\mathrm {div}\frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}} \in {\mathbb {L}}^2$ for a.a. $(\omega , t) \in \Omega \times (0,T)$. Indeed, from [4,  Lemma 3.2] for $\delta > 0$, $n< \infty$ we deduce by parabolic regularity theory that $X^{\varepsilon ,\delta }_{2,n}(\omega , t) \in {\mathbb {H}}^2$ for a.a. $(\omega , t) \in \Omega \times (0,T)$ and a direct calculation yields that

$$\begin{aligned} \left| \mathrm {div}\frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}}\right| \le 2\frac{|\nabla X^{\varepsilon ,\delta }_{2,n}||\nabla ^2 X^{\varepsilon ,\delta }_{2,n}|}{\big (|\nabla X^{\varepsilon ,\delta }_{2,n}|^2 + \varepsilon ^2\big )^{\frac{3}{2}}} + \frac{|\Delta X^{\varepsilon ,\delta }_{2,n}|}{\sqrt{|\nabla X^{\varepsilon ,\delta }_{2,n}|^2 + \varepsilon ^2}}. \end{aligned}$$

We show the inequality in (5) by the integration by parts formula using a density argument. We fix $(\omega , t)\in \Omega \times (0,T)$ and proceed below with $X_1^{\varepsilon } \equiv X_1^{\varepsilon }(\omega ,t)$, $X^{\varepsilon ,\delta }_{2,n}\equiv X^{\varepsilon ,\delta }_{2,n}(\omega ,t)$. We consider an approximating sequence $x_k \in C^{\infty }({\mathcal {O}})\cap BV({\mathcal {O}})$, s.t. $x_k \rightarrow X_1^{\varepsilon }$ strongly in ${\mathbb {L}}^1$ and

$$\begin{aligned} {\mathcal {J}}_{\varepsilon ,{0}}(x_k) \rightarrow {\mathcal {J}}_{\varepsilon ,{0}}(X_1^{\varepsilon }) \,\,{\text { for }} k \rightarrow \infty , \end{aligned}$$

(6)

cf., [1,  Theorems 10.1.2, 13.4.1 and Remark 10.2.1] or [6,  Theorem 5.2].

Note that, since $X_1^{\varepsilon }(\omega ,t)\in {\mathbb {L}}^2$ it is straightforward to modify the proof of [1,  Theorems 10.1.2] (see for instance [1,  Proposition 2.2.4] for the ${\mathbb {L}}^p$ properties of the mollifiers) such that the sequence $x_k$ converges strongly in ${\mathbb {L}}^2$:

$$\begin{aligned} \Vert x_k - X_1^{\varepsilon }\Vert _{{\mathbb {L}}^2}\rightarrow 0 \,\,{\text { for }} k \rightarrow \infty . \end{aligned}$$

(7)

Using the integration by parts formula [1,  Theorem 10.2.1] we obtain that

$$\begin{aligned}&\left( x_k-X^{\varepsilon ,\delta }_{2,n},-\mathrm {div}\frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}} \right) =\left( \nabla (x_k-X^{\varepsilon ,\delta }_{2,n}), \frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}} \right) \nonumber \\&\quad +\int _{\partial {\mathcal {O}}} \gamma _0(x_k) \frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}}\cdot \nu \,\mathrm {d}{\mathcal {H}}^{n-1}-\int _{\partial {\mathcal {O}}} \gamma _0(X^{\varepsilon ,\delta }_{2,n}) \frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}}\cdot \nu \,\mathrm {d}{\mathcal {H}}^{n-1}, \end{aligned}$$

(8)

where $\nu$ is the outer unit normal vector to $\partial {\mathcal {O}}$ and ${\mathcal {H}}^{n-1}$ is the Hausdorff measure on $\partial {\mathcal {O}}$.

Since $X^{\varepsilon ,\delta }_{2,n}\in {\mathbb {H}}^1_0$ it holds that $\gamma _0(X^{\varepsilon ,\delta }_{2,n})=0$ and the second boundary integral vanishes. The first boundary integral can be estimated as

$$\begin{aligned} \int _{\partial {\mathcal {O}}}&\gamma _0(x_k) \frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}}\cdot \nu \,\mathrm {d}{\mathcal {H}}^{n-1} \le \int _{\partial {\mathcal {O}}} \left| \gamma _0(x_k)\right| \left| \frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}}\cdot \nu \right| \ d{\mathcal {H}}^{n-1} \nonumber \\&\le \int _{\partial {\mathcal {O}}} \left| \gamma _0(x_k)\right| \,\mathrm {d}{\mathcal {H}}^{n-1} = {\int _{\partial {\mathcal {O}}} \left| \gamma _0(X_1^{\varepsilon })\right| \,\mathrm {d}{\mathcal {H}}^{n-1}}, \end{aligned}$$

(9)

where the last equality follows from the fact that the trace of $x_k \in C^{\infty }({\mathcal {O}})\cap BV({\mathcal {O}})$ coincides with the trace of $X_1^{\varepsilon }$, cf. [1,  Remark 10.2.1].

By the convexity of ${\mathcal {J}}_{\varepsilon ,0}$ we deduce that

$$\begin{aligned} \left( \nabla (x_k-X^{\varepsilon ,\delta }_{2,n}), \frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}} \right) \le {\mathcal {J}}_{\varepsilon ,0}(x_k)-{\mathcal {J}}_{\varepsilon ,0}(X^{\varepsilon ,\delta }_{2,n}). \end{aligned}$$

(10)

Hence, (5) follows after substituting (10), (9) into (8) and taking the limit for $k\rightarrow \infty$ and noting (6), (7).

The rest of the proof follows analogously to the original proof of [4,  Theorem 3.1].

4 Convergence of the full discretization

In the proof of [4,  Lemma 4.4] it is concluded that

$$\begin{aligned}&\frac{1}{2}\sum _{K,K'\in {\mathcal {T}}_h} {\bar{v}}_h^TA_K^TM^{-1}A_{K'}{\bar{v}}_h \left( (\vert \nabla v_h\vert ^2+\varepsilon ^2)_{K}^{-\frac{1}{2}}+(\vert \nabla v_h\vert ^2+\varepsilon ^2)_{K'}^{-\frac{1}{2}} \right) \\&\quad \ge \frac{1}{2}\sum _{K,K' \in {\mathcal {T}}_h}\sqrt{(\vert \nabla v_h\vert ^2+\varepsilon ^2)_{K'}^{-\frac{1}{2}}}{\bar{v}}_h^TA_K^TM^{-1}A_{K'}{\bar{v}}_h \sqrt{(\vert \nabla v_h\vert ^2+\varepsilon ^2)_K^{-\frac{1}{2}}} \ge 0, \end{aligned}$$

which is not justified. Lemma 4.4 is required to obtain the estimate (48) in [4,  Lemma 4.5] (note that the continuos counterpart of the estimate in Lemma 3.2 is obtained using Proposition 2.1), which is in turn required to show [4,  Theorem 4.1].

In this section we show an analogue of the estimate in [4,  Lemma 4.4] for a slightly modified numerical scheme in dimension $d=1$. Given $J \in {\mathbb {N}}$ and a mesh size $h=1/J$ we consider a uniform partiton ${\mathcal {T}}_h=\cup _{j=1}^J T_j$ of the spatial domain ${\mathcal {O}}=(0,1)$ into subintervals $T_j=(x_{i-1},x_i)$ with nodes $x_j=jh$, $j=0,\dots , J$. As in [4] we consider a finite element space ${\mathbb {V}}_h\subset {\mathbb {H}}^1_0$ of piecewise linear globally continuous functions on subordinated to ${\mathcal {T}}_h$. The standard nodal interpolation operator ${\mathcal {I}}_h:C({\bar{{\mathcal {O}}}})\rightarrow {\mathbb {V}}_h$ is defined as

$$\begin{aligned} {\mathcal {I}}_h \Phi (x_j)=\Phi (x_j) \quad \forall \; j=0,\ldots , L. \end{aligned}$$

We define the discrete (mass-lumped) ${\mathbb {L}}^2$-inner product $\left( \cdot ,\cdot \right) _h$ on ${\mathbb {V}}_h$ as

$$\begin{aligned} \left( \varphi ,\psi \right) _h&=\int \limits _{{\mathcal {O}}}{\mathcal {I}}_h(\langle \varphi , \psi \rangle )(x)\,\mathrm {d}x=h \sum _{j=1}^{J-1} \varphi (x_j),\psi (x_j) ~~ {\text {for}}\,\, \varphi ,\psi \in {\mathbb {V}}_h, \end{aligned}$$

(11)

with the corresponding discrete norm $\Vert \psi \Vert ^2_h=\left( \psi ,\psi \right) _h$.

It is well known that the above discrete inner product and the norm satisfy (cf. [5]):

$$\begin{aligned} \Vert v_h\Vert _{{\mathbb {L}}^2} \le \Vert v_h\Vert _h&\le C\Vert v_h\Vert _{{\mathbb {L}}^2}~~&\forall \; v_h \in {\mathbb {V}}_h, \end{aligned}$$

(12)

$$\begin{aligned} \left| \left( v_h,w_h \right) _h-\left( v_h,w_h \right) \right|&\le Ch \Vert v_h\Vert _{{\mathbb {L}}^2}\Vert w_h\Vert _{{\mathbb {H}}^1}~~&\forall \; v_h,w_h \in {\mathbb {V}}_h. \end{aligned}$$

(13)

We define the mass-lumped Discrete Laplace operator $\Delta _h: {\mathbb {V}}_h \rightarrow {\mathbb {V}}_h$ through the identity

$$\begin{aligned} \left( \Delta _h v_h, w_h \right) _h&=- \left( \nabla v_h,\nabla w_h \right) . \end{aligned}$$

(14)

The next lemma is the counterpart of [4,  Lemma 4.4] for the 1d discrete Laplace operator (14). Numerical experiments (not stated in this paper) indicate that the result also holds for $d>1$ (possibly under some additional assumptions on the shape of the mesh). Nevertheless, the proof of the result for $d>1$ remains open, so far.

Lemma 3.1

Let $\Delta _h$ be the discrete Laplacian defined by (14). Then for any $v_h \in {\mathbb {V}}_h$, $\varepsilon , h > 0$ the following inequality holds:

$$\begin{aligned} \left( \frac{\nabla v_h}{ \sqrt{ \vert \nabla v_h \vert ^2 +\varepsilon ^2}},\nabla (-\Delta _h v_h) \right) \ge 0. \end{aligned}$$

Proof

Since ${\mathbb {V}}_h$ is the space of piecewise linear functions over ${\mathcal {T}}_h$, it holds for $v_h \in {\mathbb {V}}_h$ that

$$\begin{aligned} \delta _x v^j_h := \partial _x v_h(x) \big |_{T_j}=\frac{v_h(x_j)-v_h(x_j)}{h}. \end{aligned}$$

By definition (11) and (14) we deduce that

$$\begin{aligned} \Delta _h v_h^j := \Delta _h v_h(x_j)= \frac{v_h(x_{j+1})-2v_h(x_{j})+v_h(x_{j-1})}{h^2}=\frac{\delta _x v^{j+1}_h-\delta _x v^{j}_h}{h}, \end{aligned}$$

and $\Delta _h v_h^0 = \Delta _h v_h^J = 0$.

By the above properties we deduce that

$$\begin{aligned}&\left( \frac{\nabla v_h}{ \sqrt{ \vert \nabla v_h \vert ^2 +\varepsilon ^2}},\nabla (-\Delta _h v_h) \right) =-\left( \frac{\partial _x v_h}{\sqrt{\vert \partial _x v_h\vert ^2+\varepsilon ^2}},\partial _x \Delta _h v_h \right) \\&\quad =-\sum _{j=1}^{J}\int _{T_j}\frac{\partial _x v_h}{\sqrt{\vert \partial _x v_h\vert ^2+\varepsilon ^2}}\partial _x \Delta _h v_h\,\mathrm {d}x \\&\quad =-h\sum _{j=1}^{J}\frac{\delta _x v^j_h}{\sqrt{\vert \delta _x v^j_h\vert ^2+\varepsilon ^2}}\delta _x \Delta _h v_h^j =-\sum _{j=1}^{J}\frac{\delta _x v^j_h}{\sqrt{\vert \delta _x v^j_h\vert ^2+\varepsilon ^2}} \left( \Delta _h v_h^j-\Delta _h v_h^{j-1}\right) \\&\quad =-\sum _{j=1}^{J}\frac{\delta _x v^j_h}{\sqrt{\vert \delta _x v^j_h\vert ^2+\varepsilon ^2}} \Delta _h v_h^j +\sum _{j=1}^{J}\frac{\delta _x v^j_h}{\sqrt{\vert \delta _x v^j_h\vert ^2+\varepsilon ^2}} \Delta _h v_h^{j-1} \\&\quad =-\sum _{j=1}^{J-1}\frac{\delta _x v^j_h}{\sqrt{\vert \delta _x v^j_h\vert ^2+\varepsilon ^2}} \Delta _h v_h^j+\sum _{j=1}^{J-1}\frac{\delta _x v^{j+1}_h}{\sqrt{\vert \delta _x v^{j+1}_h\vert ^2+\varepsilon ^2}} \Delta _h v_h^j \\&\quad =\sum _{j=1}^{J-1}\left( \frac{\delta _x v^{j+1}_h}{\sqrt{\vert \delta _x v^{j+1}_h\vert ^2+\varepsilon ^2}}-\frac{\delta _x v^j_h}{\sqrt{\vert \delta _x v^j_h\vert ^2+\varepsilon ^2}} \right) \Delta _h v_h^j \\&\quad =\frac{1}{h}\sum _{j=1}^{J-1}\left( \frac{\delta _x v^{j+1}_h}{\sqrt{\vert \delta _x v^{j+1}_h\vert ^2+\varepsilon ^2}}-\frac{\delta _x v^j_h}{\sqrt{\vert \delta _x v^j_h\vert ^2+\varepsilon ^2}} \right) (\delta _x v_h^{j+1}-{\delta _x v_h^{j}}) \\&\quad \ge 0 \end{aligned}$$

where we used the convexity of $\sqrt{|\cdot |^2+\varepsilon ^2}$ to deduce the last inequality. $\square$

Using the above lemma one can show the convergence for a slight modification of the fully discrete numerical scheme of [4] where the standard ${\mathbb {L}}^2$-inner product is replaced by the discrete inner product (11) as follows: given $x_0,\, g \in {\mathbb {L}}^2$ we set $X^{0}_{\varepsilon ,h}={\mathcal {P}}_h x_0$, $g^h:={\mathcal {P}}_h g$ and obtain $X^{i}_{\varepsilon ,h}$ for $i=1,\dots , N$ as the solution of the following system:

$$\begin{aligned} \left( X^{i}_{\varepsilon ,h},v_h \right) _h =&\left( X^{i-1}_{\varepsilon ,h},v \right) _h-\tau \left( \frac{\nabla X^{i}_{\varepsilon ,h}}{ \sqrt{ \vert \nabla X^{i}_{\varepsilon ,h} \vert ^2 +\varepsilon ^2}},\nabla v_h \right) \nonumber \\&-\tau \lambda \left( X^{i}_{\varepsilon ,h} -g^h,v_h \right) _h+\left( X^{i-1}_{\varepsilon ,h},v_h \right) _h\Delta _i W&\forall \; v_h \in {\mathbb {V}}_h. \end{aligned}$$

(15)

By the equivalence of the norms $\Vert \cdot \Vert$ and $\Vert \cdot \Vert _h$ (cf. (12)) the convergence of the above numerical approximation for $d=1$ follows as in [4] with [4,  Lemma 4.4] replaced by Lemma 3.1.

We note that the convergence proof remains valid for $d\ge 1$ in the case of the time-semi discrete variant of the original numerical scheme from [4]:

$$\begin{aligned} \left( X^{i}_{\varepsilon },\varphi \right) =&\left( X^{i-1}_{\varepsilon },\varphi \right) -\tau \left( \frac{\nabla X^{i}_{\varepsilon }}{ \sqrt{ \vert \nabla X^{i}_{\varepsilon } \vert ^2 +\varepsilon ^2}},\nabla \varphi \right) \\&-\tau \lambda \left( X^{i}_{\varepsilon } -g,\varphi \right) +\left( X^{i-1}_{\varepsilon },\varphi \right) \Delta _i W&\forall \; \varphi \in {\mathbb {H}}^1_0. \end{aligned}$$

In the semi-discrete setting one employs the continuous counterpart of Lemma 3.1 and proceeds as in the proof of [4,  Lemma 3.2] to obtain the space-continuous version of the stronger estimate (48) in Lemma 4.5 from [4]. Then the convergence proof of the above semi-discrete numerical scheme follows analogically as in the case of the fully discrete numerical approximation; we skip the detailed exposition for brevity and instead refer to [3,  Section 4], from where the necessary components of the proof can be deduced.

Finally, we conclude that a convergence proof of the fully discrete numerical approximation for $d \ge 1$, which avoids the use of [4,  Lemma 4.4], is provided in the upcoming paper [3].