An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation
Xiaojie Wang




Full discrete scheme for stochastic Allen-Cahn equation
T1This work was supported by NSF of China (11671405, 11971488, 91630312), Natural
Science Foundation of Hunan Province (2020JJ2040, 2018JJ3628).


The author wants to thank the Tianyuan Mathematical Center in Northeast China
for the hospitality 


when this work was presented in a conference in June of 2018, hosted by the center.


The author  also  wants to thank Yuying Zhao for her kind help with excellent typesetting
and Ruisheng Qi and Meng Cai for their useful comments based on carefully reading the manuscript.












School of Mathematics and Statistics,

Central South University, Changsha, Hunan, China

e1


E-mail: 



X. Wang






In [Becker and Jentzen, Stoch. Proc. Appl. 129:28–69, 2019] and [Becker, et al., arXiv preprint arXiv:1711.02423, 2017],  
an explicit temporal semi-discretization scheme and a space-time full-discretization scheme were, respectively, introduced and analyzed for 
the additive noise-driven stochastic Allen-Cahn type equations, with strong convergence rates recovered.
The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen-Cahn equation 
with cubic nonlinearity, perturbed by additive space-time white noise. The approximation is easily implementable, performing the spatial 
discretization by a spectral Galerkin method and the temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator 
scheme. Error bounds in a strong sense are analyzed for both the spatial semi-discretization and the spatio-temporal full discretization, 
with convergence rates in both space and time explicitly identified. It turns out that the obtained convergence rate of 
the new scheme is, in the temporal direction, twice as high as existing ones in the literature. 
Numerical results are finally reported to confirm the previous theoretical findings.


[class=MSC]



65C30, 60H35,  60H15



stochastic Allen-Cahn equation
cubic nonlinearity
spectral Galerkin method
tamed exponential integrator scheme
strong convergence rate






§ INTRODUCTION



As an active area of research, numerical study of evolutionary stochastic partial differential equations (SPDEs) 
has attracted increasing attention in the past decades 

(see, e.g.,  monographs <cit.> and references therein). 

Albeit much progress has been made,  it is still not well-understood, especially for numerical analysis of 
SPDEs with non-globally Lipschitz nonlinearities. 

The present work attempts to make a contribution in this direction and examine

a space-time full-discretization scheme for a typical example of parabolic SPDEs with super-linearly growing nonlinearity, 
i.e., the stochastic Allen-Cahn equation.



The driving noise is a space-time white one, which is of special interest as it can best model the fluctuations 
generated by microscopic effects in a homogeneous physical system, including, for example, 
molecular collisions in gases and liquids, electric fluctuations in resistors <cit.>.
A lot of researchers carried out numerical analysis of SPDEs subject to such noise, e.g., 
<cit.>,      
to just mention a few.

Numerically solving the continuous problem on a computer forces us to perform 
both spatial and temporal discretizations. In space, we discretize SPDE (<ref>) 
by a spectral Galerkin method,
resulting in a system of finite dimensional stochastic differential equations (SDEs). Based on the spatial discretization,  
we propose a nonlinearity-tamed accelerated exponential time-stepping scheme given by (<ref>). 

The resulting approximation errors of both the spatial discretization and the space-time fully discrete scheme are carefully analyzed, 
with strong convergence rates successfully recovered.











More accurately, by X(t_m) we denote the unique mild solution of the underlying  SPDE  taking values at temporal grid points 
t_m = m τ, m ∈{ 0, 1,..., M } with uniform time step-size τ = TM > 0 and by Y^M,N_t_m the numerical approximations 
of X(t_m),  produced by the proposed fully discrete scheme. 
The approximation error measured in L^p ( Ω; H), p ∈ [2, ∞) reads
(cf. Theorem <ref>):

    sup_ 0 ≤ m ≤ M X ( t_m ) -  Y^M,N_t_m_L^p ( Ω; H) ≤
    C 
     ( N^ - β + τ ^β), 
      ∀ β∈ (0, 12).

Here H := L^2 ((0, 1);  ) and the constant C depends on p, β, T and the initial value of the SPDE,
but does not depend on the discretization parameters M, N. 




Over the last two years, several research works were reported on numerical analysis of space-time white noise driven SPDEs with 
cubic (polynomial) nonlinearity <cit.>.
Becker and Jentzen <cit.> in 2016 introduced two nonlinearity-truncated Euler-type approximations
for pure time discretizations of stochastic Ginzburg Landau type equations 
with slightly more general polynomials. There a strong convergence rate of order almost 1/4 is identified. 
More recently when the first preprint of this work was almost finished, we were aware of four other preprints 
<cit.> submitted to arXiv, 
concerning with numerical approximations of 

similar SPDEs.
Becker, Gess, Jentzen and Kloeden <cit.> 
proposed new types of truncated exponential Euler space-time fully discrete schemes for the same problem 
as in  <cit.> and derived strong convergence rates of order almost 1/2 in space  
and order almost 1/4 in time. 
Later, Bréhier and Goudenège <cit.> and Bréhier, Cui and Hong <cit.> analyzed 
some splitting time discretization schemes and obtained strong convergence rates of order 1/4. 


Liu and Qiao <cit.> investigated a spectral Galerkin backward implicit Euler full discretization,
with strong convergence rates of order almost 1/2 in space  and order 1/4 in time achieved.
As clearly implied by (<ref>), the spatial convergence rate coincides with those in 
<cit.>, but the strong convergence rate of our time-stepping scheme 
can be of order almost 1/2, essentially twice as high as those in  
<cit.>. 
Despite getting involved with linear functionals of the noise process, 
the newly proposed scheme  is explicit, 
easily implementable and does not cost additional computational efforts
(see comments in section <ref> for the implementation of the linear functionals of the noise process).


It is important to emphasize that, proving the error estimate (<ref>) is challenging,
confronted with two essential difficulties, one being to derive uniform a priori moment bounds for the numerical approximations
with super-linearly growing nonlinearity (see relevant comments in <cit.>) 
and the other to recover the temporal convergence rate of order almost 12, 
instead of order (almost) 14 in the existing literature.  With regard to the former, we essentially rely on certain estimates for 
deterministic perturbed PDEs (<ref>), as elaborated in subsection <ref>.
The uniform a priori moment L_∞-bounds are derived based on a certain bootstrap argument, by showing [ _Ω_ R^τ,t_m Y^M,N_t_m_V^p] < ∞ and [
   _Ω^c_R^τ,t_mY_t_m^M,N_V^p
    ] < ∞,
   V : = C ( (0, 1),  ),
for  subevents Ω_ R^τ,t_m  with R^τ depending on τ carefully chosen 
(see subsection <ref>).
The latter difficulty is addressed by fully exploiting






















the global monotonicity condition on the nonlinearity 

in conjunction with smoothing property of the analytic semigroup, commutativity properties of the nonlinearity and
improved temporal Hölder regularity results in negative Sobolev spaces (consult subsection <ref>
and particularly the treatment of J_1 in the proof of Theorem <ref> for details).



Furthermore, we would like to point out that the improvement of convergence rate is essentially 
credited to fully preserving the stochastic convolution in the time-stepping scheme (<ref>).

Such a kind of accelerating technique was used by Jentzen and Kloeden <cit.>,
to solve nearly linear parabolic SPDEs 
and has been further examined and extended in different settings <cit.>, where 

a globally Lipschitz condition imposed on nonlinearity is indispensable in the error analysis.




When the nonlinearity grows super-linearly and the globally Lipschitz condition is thus violated,  one can in general not expect
the usual accelerated exponential time-stepping schemes converge in the strong sense,
based on the observations in 

<cit.>.  To address this issue,  
we introduce a taming technique previously used in <cit.> 
for ordinary SDEs, and propose a nonlinearity-tamed version of accelerated exponential Euler scheme for the time discretization.

Although the idea of constructing the explicit scheme in this paper is inspired by the aforementioned works, 
the approach of the error analysis for the full discretization of the stochastic Allen-Cahn equation with space-time white noise is original
and much more involved than that in both the finite dimensional non-globally Lipschitz SDE setting and the globally Lipschitz SPDE 
setting (see section <ref>).
Moreover, our approach is much easier than that in <cit.>, 
also treating explicit full-discrete schemes for Allen-Cahn type SPDEs.






Finally, we mention that, just one spatial dimension is considered here because 
the space-time white noise driven SPDE only allows for a mild solution 
with a positive (but very low) order of regularity in one spatial dimension.
It is because of the low order of regularity that the error analysis becomes difficult.
Further, the error analysis of the space-time full discretization is significantly more involved than 
that of the pure time semi-discretization (compare <cit.> and <cit.> 
and see comments therein).

Strong convergence analysis of numerical methods for smoother noise (e.g., trace-class noise) driven 
stochastic Allen-Cahn equation in multiple spatial dimensions has been done in our recent work <cit.>
and further strong and weak approximations of parabolic SPDEs with non-globally Lipschitz nonlinearity 
will be our forthcoming works

(see also, e.g.,




<cit.>
for revalent topics).






































































The rest of this paper is organized as follows. In the next section we collect some basic facts and
present the well-posedness of the stochastic problem under given assumptions. 
Section <ref> and Section <ref> are, respectively, 
devoted to the analysis of strong convergence rates for both the spatial semi-discretization and  
the spatio-temporal full discretization of the underlying SPDEs.   
Numerical results are included in section <ref>  to test previous theoretical findings.






























§ WELL-POSEDNESS OF THE STOCHASTIC PROBLEM











Throughout this article, we are interested in the additive space-time white noise driven 
stochastic Allen-Cahn equation with cubic nonlinearity, described by

    {[ ∂ u /∂ t (t,x) = ∂^2 u /∂ x^2  (t,x) + f(u(t,x)) + Ẇ (t, x),    x ∈ D,
          t ∈ (0, T],;                                                               u(0, x) = u_0(x),   x ∈ D,;                                                   u(t, 0) = u (t, 1) = 0,    t ∈ (0, T]. ].

Here D := (0, 1),  T > 0, f ℝ→ℝ is given by 
f (v) =  a_3 v^3  + a_2 v^2 + a_1 v + a_0,  
a_3 < 0,   a_2, a_1, a_0, v ∈,
and Ẇ (t, · ) stands for a formal time derivative of a cylindrical I-Wiener process <cit.>.



In order to define a mild solution of (<ref>) following the semigroup approach in <cit.>,
we attempt to put everything into an abstract framework.
Given a real separable Hilbert space (H, ⟨·, ·⟩, · ) with · = ⟨·, ·⟩^1/2,
by ℒ(H) we denote the space of bounded linear operators from 
H to H endowed with the usual operator norm ·_ℒ(H).
Additionally, we denote by ℒ_2(H) ⊂ℒ(H) the subspace
consisting of all Hilbert-Schmidt operators from H to H <cit.>. 
It is known that ℒ_2(H) 
is a separable Hilbert space, equipped with the scalar product
⟨Γ_1, Γ_2  ⟩_ℒ_2(H) 
: = 
∑_n ∈⟨Γ_1 η_n, Γ_2 η_n  ⟩,
and norm
Γ_ℒ_2(H)  
:=
 (
∑_n ∈Γη_n ^2
 )^1/2,
independent of the particular choice of orthogonal basis {η_n}_n ∈ of H.
Below we sometimes write ℒ_2 := ℒ_2(H) for brevity.
If Γ∈ℒ(H) and Γ_1, Γ_2 ∈ℒ_2(H), then


| ⟨Γ_1, Γ_2  ⟩_ℒ_2(H) | 
≤Γ_1 _ℒ_2(H)Γ_2  _ℒ_2(H),

ΓΓ_1_ℒ_2(H)≤Γ_ℒ(H)Γ_1_ℒ_2(H).


By L^γ ( D;  ), γ≥ 1 (L^γ ( D ) for short) 
we denote a Banach space consisting of γ-times integrable functions 

and by V : = C ( D,  ) a Banach space of continuous functions with usual norms.













To reformulate (<ref>) as an abstract problem,  we make the following assumptions.
[Linear operator A]
Denote D : = (0, 1) and let H = L^2 (D;  )  be a real separable Hilbert space, 
equipped with usual product  ⟨·, ·⟩
and norm · = ⟨·, ·⟩^1/2.
Let - A dom (A) ⊂ H → H be the Laplacian with homogeneous Dirichlet boundary conditions, 
defined by - A u = Δ u, u ∈dom(A) := H^2 ∩ H_0^1.



The above setting assures that there exists an increasing sequence of real numbers λ_i=π^2 i^2, i ∈ℕ 
and an orthonormal basis {e_i ( x ) =√(2)sin(i π x),  x
∈(0,1)}_i ∈ℕ such that A e_i = λ_i e_i.
In particular, the linear unbounded operator A is positive, i.e., ⟨ - A v, v ⟩≤ - λ_1  v ^2,
for all v ∈dom(A).
Moreover, -A generates an analytic semigroup E(t) = e^-t A, t ≥ 0 on H







and we can define the fractional powers of A, i.e., A^γ, γ∈ℝ 
and the Hilbert space Ḣ^γ :=dom(A^γ/2), equipped with inner product  
⟨·, ·⟩_γ : = ⟨ A^γ/2·, 
A^γ/2·⟩ 
and norm ·_γ = ⟨·, ·⟩_γ^1/2 
<cit.>. Moreover, Ḣ^0 = H and 
Ḣ^γ⊂Ḣ^δ,  γ≥δ. It is well-known that <cit.>, 
for a positive constant c >0,

    A^γ E(t)_ℒ(H)≤    c t^-γ,    t >0, γ≥ 0,
    
     A^-ρ (I-E(t))_ℒ(H)≤    c t^ρ,    t >0, ρ∈ [0,1],















and accordingly one can verify that <cit.>


    ∫_s^t  A^ρ/2 E ( t - r ) u ^2   r 
    ≤
    c 
    ( t -  s )^ 1 - ρ u ^2,
      ∀ u ∈ H, 
     ρ∈ [0, 1],
     
    0≤ s ≤ t.

Here the constant c, might depending on γ, can be chosen to be independent of ρ∈ [0, 1].
Also, it is evident to see that

    A^β-1/2_ℒ_2 (H) 
    =
    π^β - 1(
    ∑_i ∈ i^ 2 ( β - 1) )^1/2
    < ∞,    for any β < 12.



[Nonlinearity]
Let F  L^6 ( D; ) → H be  a deterministic mapping defined by






    F (v) ( x ) 
    =
    f ( v ( x ) )
    := a_3 v ^3(x)   + a_2 v ^2(x)  + a_1 v (x) + a_0,  
    
    x ∈ (0, 1),
    a_3 < 0,
    a_2, a_1, a_0 ∈,
     
    v ∈ L^6 ( D; )
    .


It is easy to find constants L_0, L_1 ∈ (0, ∞) such that

    ⟨ u - v,  F (u) - F (v) ⟩   ≤ L_0  u - v ^2,    u , v ∈ V,
    
     F (u) - F (v)    ≤ L_1 ( 1 +  u _V^2 +   v _V^2 )  u - v ,     u, v ∈ V.

The second property in (<ref>) immediately implies, for some L_2 ∈ (0, ∞),

    F (u) ≤ L_1 ( 1 +  u _V^2 )  u   +  F ( 0 ) ≤ L_2 ( 1 +  u _V^3 ),     u ∈ V.

Evidently, the constants L_0, L_1, L_2 only depend on coefficients of f, i.e., a_0, a_1, a_2, a_3.

[Noise process]
Let { W (t) }_t ∈[0, T] be a cylindrical I-Wiener process on a 
probability space  (Ω,ℱ,ℙ) 
with a normal filtration {ℱ_t}_ t∈ [0, T], represented by a formal series,

    W (t) := ∑_n = 1^∞β_n ( t ) e_n,
      
    t ∈ [0, T],

where {β_n ( t ) }_n ∈, t ∈ [0, T] is a sequence of independent real-valued standard Brownian motions 
and {e_n = √(2)sin(n π x),  x ∈(0,1) }_n ∈ is a complete orthonormal basis  of H.
























[Initial value] 
Let the initial data X_0 Ω→  H, given by X_0 ( · ) = u_0 ( · ),
be an ℱ_0/ℬ(H)-measurable random variable.
For sufficiently large positive number p_0 ∈ and for any β < 12, 

there exists constants K_β depending on β, p_0 
and K̂_V only depending on p_0 such that



    X_0 _L^p_0(Ω, Ḣ^β )≤ K_β < ∞,
    
    
         X_0 _L^p_0(Ω, V )≤K̂_V < ∞.









At the moment, we are prepared to formulate the concrete problem (<ref>) as 
an abstract stochastic evolution equation in the Hilbert space H, 

    {[ X(t) + A X(t)  t = F ( X(t) )   t +   W(t),     t ∈ (0, T],;                                                 X(0) = X_0, ].

where X (t, · ) = u (t, ·) and the abstract items A, F, X_0 
are defined in Assumptions <ref>-<ref>.

The above assumptions suffice to establish the well-posedness and regularity results of the mild solution 
to  (<ref>), defined by (<ref>) later. Before that, we provide some 
regularity properties of the stochastic convolution.



Let Assumptions <ref>, <ref> be fulfilled.
For any p ∈ [ 2, ∞ )
there exists a constant C_0 dependent of p but independent of β∈ [0, 12)  such that 
the stochastic convolution {𝒪_t }_t ∈ [0, T] satisfies

    𝒪_t  _L^p(Ω, Ḣ^β )   ≤
    C_0
    A^β-1/2_ℒ_2 (H)  
    < ∞,
        𝒪_t : = ∫_0^t E(t-s)  W(s),
    
    
    
    
    
    
    
    
    𝒪_t - 𝒪_s _L^p(Ω, H)   ≤
    C_0 
    A^β-1/2_ℒ_2 (H) 
    (t -s)^β/2 , 
       0 ≤ s < t ≤ T.

Moreover, we have

    [ sup_t ∈ [0, T] 𝒪_t _V^p   ]   < ∞.



Recalling (<ref>) one can validate the first two estimates easily, see, e.g., <cit.>.
The last assertion can, e.g., be found in <cit.> and <cit.>.
Owing to the above regularity properties of the stochastic convolution, we can get the corresponding regularity properties of the
mild solution to (<ref>) as follows.

 
Under Assumptions <ref>-<ref>, SPDE (<ref>) possesses a unique mild solution 
X: [0,T] ×Ω→ V with continuous sample paths,

determined by, 

    X(t)
        =
        E(t) X_0
        +
        ∫_0^t E(t-s) F ( X( s ) )   s
        +
        𝒪_t
      ℙ
    .





For any p ∈ [2, ∞) and β < 12, there exists positive constants C_1, C_2 
depending on p, T, {a_i}_i=0^3  but not  depending on β such that,

    sup_ t ∈ [0, T]  X ( t ) _L^p(Ω, V )   ≤ 
            C_1 ( 1 +  X_0 _L^p(Ω, V )),
    
    sup_ t ∈ [0, T]  X ( t ) _L^p(Ω, Ḣ^β )   ≤ 
           C_2 ( 1 +  X_0 _L^  p (Ω, Ḣ^β )  +  X_0 _L^  3p (Ω, V ) ^3 
             +
             A^β-1/2_ℒ_2 ( H ) ).

Moreover, there exists a constant C_3 ∈ [0, ∞) 
depending on p, T, {a_i}_i=0^3 but not  depending on β such that,
for any β < 12 and 0 ≤ s < t ≤ T,

    X ( t ) - X ( s ) _L^p(Ω, H)≤ C_3 
    ( 1 +  X_0 _L^  p (Ω, Ḣ^β )  +  X_0 _L^  3p (Ω, V ) ^3 
             +
             A^β-1/2_ℒ_2 ( H ) )
    
    (t -s)^β/2.



The existence of the unique mild solution and the regularity assertion (<ref>) are based on 
<cit.> and (<ref>).
The rest of estimates in Theorem <ref> can be verified by standard arguments 
(consult, e.g., <cit.>).





§ SPATIAL SEMI-DISCRETIZATION
 


This section concerns the error analysis for a spectral Galerkin spatial semi-discretization 
of the underlying problem (<ref>).

For N∈ℕ we define a finite dimensional subspace of H by


    H^N := {e_1, e_2, ⋯, e_N },


and the projection operator P_N Ḣ^α→ H^N by
P_N ξ = ∑_i=1^N ⟨ξ, e_i ⟩ e_i, 
  
  ∀ ξ∈Ḣ^α,  α∈.
Here H^N is chosen as the linear space spanned by the N first eigenvectors of 
the dominant linear operator A. It is not difficult to deduce that

    ( P_N - I ) φ≤λ_N+1^- α/2φ_α≤ N^ - αφ_α, 
      ∀ φ∈Ḣ^α,  α≥0.

Additionally, define A_N  H → H^N as
A_N = A P_N, which generates an analytic semigroup E_N(t) = e^-t A_N, t ∈ [0, ∞) in H^N. 
Then the spectral Galerkin approximation of (<ref>) results in the following finite dimensional SDEs,

    {[ X^N(t) + A_N X^N ( t )   t =  F_N ( X^N (t) )   t  +  P_N    W (t), 
           t ∈ (0, T],;                                                                       X^N(0) = P_N X_0, ].

where we write F_N : = P_N F for short. It is clear to see that (<ref>) admits a unique solution in H^N.
By the variation of constant, the corresponding solution can be written as

    X^N(t)
        =
        E_N(t) P_N X_0
        +
        ∫_0^t E_N (t - s ) P_N F ( X^N( s ) )   s
        +
        ∫_0^t E_N(t-s) P_N   W (s),  ℙ.

















In the error analysis for the spatial semi-discretization (<ref>), 
we require P_N X ( t ) _ L^p(Ω, V ) , t > 0 to be
uniformly bounded with respect to N ∈, which can be deduced based on the following two auxiliary results. 
The first one is a direct consequence of <cit.> with t_1 = 0 and 
we follow the same arguments there to provide a short proof here.
 
 

Let {𝒪_t }_t ∈ [0, T] be the stochastic convolution defined by (<ref>)
and let Assumptions <ref>, <ref> be fulfilled. Then
for any p ∈ [2, ∞ ) it holds

    sup_ t ∈ [0, T], N ∈𝒪_t^N _L^p(Ω, V )  
    < ∞,
          𝒪_t^N := P_N 𝒪_t.





 Proof of Lemma <ref>.

Owing to Assumptions <ref>, <ref>, we perform the expansion of the stochastic convolution O_t^N 
and use the Itô isometry to get, for any N ∈, t ∈ [0, T], x, y ∈ D,

    [| 𝒪_t^N ( x )
       -
       𝒪_t^N ( y )|^2]
          ≤∑_1 ≤ i ≤ N[
       |
       ∫_0^t e^-λ_i(t- s )β_i(s)
       |^2
       ]
       |e_i(x)-e_i(y)|^2
       
       ≤∑_i∈
       ( 2 λ_i )^-1
       ( √(2)π i )^4/5 | x - y |^4/5(
       |e_i(x)|+|e_i(y)|
       )^6/5
       ≤
       2^6/5π^-6/5
       | x - y|^4/5∑_i∈
       i^-6/5
       ≤ 3
       | x - y|^4/5.


In the same manner, one can acquire, for any N ∈, t ∈ [0, T],

    sup_x ∈ D[
    | 𝒪_t^N ( x ) |^2
    ]
    ≤ 2.


Using the Sobolev embedding inequality W^ 15 , p⊂ V, p > 5 and 
noting that the stochastic convolution O_t^N is Gaussian one can deduce, for any N ∈, t ∈ [0, T],

    [ 𝒪_t^N_V^p ]
        ≤
       C  ∫_0^1 [ | 𝒪_t^N(x) |^p ]  x
       +
       C ∫_0^1 ∫_0^1
       [| 𝒪_t^N(x)
       -
       𝒪_t^N(y)|^p]/|x-y|^p5 + 1  x  y
    
       ≤
     C  ∫_0^1
     (
     [ | 𝒪_t^N(x) |^2 ]
     )^ p2 x
       +
       C ∫_0^1 ∫_0^1
       ([| 𝒪_t^N(x)
       -
       𝒪_t^N(y)|^2])^ p2/|x-y|^p5 +1  x  y
      
       ≤
       C (
       1
       +
       ∫_0^1 ∫_0^1 | x-y |^p5 -1 x  y
       )<∞.


Using Hölder's inequality yields the desired assertion in the case p ∈ [2, 5]. □

The second one concerns the smoothing property of the analytic semigroup.




 

Let Assumptions <ref> be fulfilled.
For any N ∈ and ψ∈Ḣ^γ,  γ∈ [0, 12 ), it holds that

    P_N E ( t ) ψ_V  ≤ 
    2^γ ( 
     5 - 4 γ 2 π ( 1 - 2 γ )  )^1/2
    t^ 2 γ -1/4ψ_γ,
      
    t > 0,
     γ∈ [0, 12 ).


 Proof of Lemma <ref>. 
Elementary facts readily yield

    P_N E ( t ) ψ_V
        =
    sup_ x ∈ [0, 1]  |
    ∑_ i = 1 ^N e^ - λ_i t  ⟨ψ, e_i  ⟩ e_i (x) |
    ≤√(2)∑_ i = 1 ^N  e^ - λ_i t   
     | 
    ⟨ψ, e_i  ⟩
     |
    
       ≤√(2)(
    ∑_ i = 1 ^N  λ_i^ - γ e^ - 2λ_i t  )^1/2(
    ∑_ i = 1 ^N
    λ_i^γ
     | 
    ⟨ψ, e_i  ⟩
     |^2
     )^1/2
    
    
    
    
    
    
    
    
    
    
    
       ≤√(2)π^ - γ(
     ∫_0^∞  x ^ - 2 γ e^ - 2 π^2 x^2 t   x
    )^1/2ψ_γ
       
    =
    2^γπ^ - 1/2 t^ 2 γ -1 / 4 (
     ∫_0^∞  y ^ - 2 γ e^ - y^2/2   y
    )^1/2ψ_γ
       ≤
    2^γ (  5 - 4 γ 2 π ( 1 - 2 γ )  )^1/2
    t^ 2 γ -1/4ψ_γ,

as required.
□



Equipped with the above two estimates, one can prove that P_N X ( t ) _ L^p(Ω, V ) , t > 0 is
uniformly bounded with respect to N ∈ as follows.


  
Let { X ( t ) }_ t ∈ [0, T] be the mild solution to (<ref>), defined by (<ref>).
Then for any p ∈ [2, ∞ ) and γ∈ [0, 12)
there exists a positive constant C,
depending on T, p, {a_i}_i=0^3, but not depending on N, β, X_0 such that


    sup_N ∈ P_N X ( t ) _ L^p(Ω, V ) ≤
    2^γ ( 
     5 - 4 γ 2 π ( 1 - 2 γ )  )^1/2 X_0 _ L^p(Ω, Ḣ^γ )  
    t^2 γ - 1/4
    +
    C 
    ( 1 +  X_0 ^3_L^3p(Ω, V ))
    
    ,
    
    
      
    t ∈ ( 0, T].



Proof of Lemma <ref>. 











Observing that
E_N(t) P_N = E (t) P_N
and using Lemmas <ref>, <ref> show

    P_N X(t) _ L^p(Ω, V ) ≤ E(t) P_N X_0 _ L^p(Ω, V ) 
        +
        ∫_0^t  E(t-s) F_N ( X( s ) ) _ L^p(Ω, V )   s
        +
         P_N 𝒪_t _ L^p(Ω, V ) 
    
    
    
    
    
    
    
    
             ≤
        2^γ ( 
     5 - 4 γ 2 π ( 1 - 2 γ )  )^1/2
        t^2 γ - 1/4 X_0 _ L^p(Ω, Ḣ^γ )  
        +
        ∫_0^t  ( t - s )^ -  1 / 4   F ( X( s ) ) _ L^p(Ω, H )   s
        +
         P_N 𝒪_t _ L^p(Ω, V ) 
         ≤
        2^γ ( 
     5 - 4 γ 2 π ( 1 - 2 γ )  )^1/2
        t^2 γ - 1/4 X_0 _ L^p(Ω, Ḣ^γ )  
        +
        43 t^3/4sup_s ∈ [0, T] F ( X( s ) ) _ L^p(Ω, H )  
        +
         P_N 𝒪_t _ L^p(Ω, V ) 
         ≤
        2^γ ( 
     5 - 4 γ 2 π ( 1 - 2 γ )  )^1/2
        t^2 γ - 1/4 X_0 _ L^p(Ω, Ḣ^γ )  
        +
        C  ( 1 + sup_s ∈ [0, T] X( s ) _ L^3p(Ω, V ) ^3  )
        +
         P_N 𝒪_t _ L^p (Ω, V ) .

Owing to (<ref>), (<ref>) and Assumption <ref>, 
one can arrive at the expected estimate.
□

Throughout this paper, 
by C we mean deterministic constants, not necessarily the same at each occurrence, 
but independent of the discretization parameters.
Now we are prepared to carry out convergence analysis for 
the spectral Galerkin discretization (<ref>).


Let Assumptions <ref>-<ref> hold. 
Let  X(t) and X^N(t) be defined through (<ref>) 
and (<ref>), respectively. Then
for any β < 12, p ∈ [ 2, ∞ ) and N ∈, 
there exists a positive constant C,
depending on T, p, {a_i}_i=0^3, but not depending on N, β, X_0 such that

    sup_ t ∈ [ 0, T ]  X(t) - X^N(t) _L^p ( Ω; H) ≤
    C 
    (
    1 
    +
      X_0 ^12_L^12p(Ω, V )
     +
      X_0 _L^  4 p (Ω, Ḣ^β∨ [ (p-1)/(2p) ]  ) ^4
     +
     A^β-1/2_ℒ_2 ^2
    )
    
    
    N^ - β.




Here and below we denote a ∨ b : = max (a, b ). 
Clearly, the above convergence rate β < 12  can be arbitrarily close to 12 
but  can not reach 12, since the quantity A^β-1/2_ℒ_2,
as calculated in (<ref>),  explodes when β tends to 12. 
This comment also applies to the full approximation error estimates in section <ref>.

Proof of Theorem <ref>.  The triangle inequality along with (<ref>),
(<ref>)  provides us that

    X(t) - X^N(t)
     _L^p( Ω, H)
    
    
    
    
    
    
    
       ≤ 
     ( I - P_N ) X(t)
     _L^p( Ω, H)
     +
      
     P_N X(t) - X^N(t)
     _L^p( Ω, H)
       ≤
    N^ - β X( t ) _L^ p  ( Ω; Ḣ^β)  
    +
      
     e^N_t
     _L^p( Ω, H)
       ≤
    C_2 ( 1 +  X_0 _L^  p (Ω, Ḣ^β )  +  X_0 _L^  3p (Ω, V ) ^3 
             +
             A^β-1/2_ℒ_2 ) 
    
    N^ - β
         
    +
      
     e^N_t
     _L^p( Ω, H)
    ,

where e^N_t : = P_N X(t) - X^N(t) = ∫_0^t E_N ( t - s )   [ F_N ( X ( s ) ) - F_N ( X^N ( s ) ) ]   s satisfies

    / t e^N_t
    =
    - A_N e^N_t  +  F_N ( X(t) ) - F_N (X^N (t) )
    =
    - A e^N_t  +  F_N ( X(t) ) - F_N (X^N (t) ).

Therefore, using (<ref>) and the Young inequality gives

    / t e^N_t ^p
    =
    p  e^N_t ^p-2⟨  e^N_t, 
    - A e^N_t  +  F ( X(t) ) - F (X^N (t) )
    ⟩
         ≤
    p  e^N_t ^p-2⟨  e^N_t, 
      F ( P_N X (t) ) - F (X^N (t) ) ⟩
    
    +
    p  e^N_t ^p-2⟨ e^N_t,   F ( X (t) ) - F (P_N X(t) ) ⟩
         ≤
    L_0  p  e^N_t ^p
    + 
    p  e^N_t ^p-1 F ( X (t) ) - F (P_N X(t) ) 
         ≤
    ( L_0 p + p -1 )   e^N_t ^p +  F ( X (t) ) - F (P_N X(t) ) ^p
    
         ≤
    ( L_0 p + p -1 )   e^N_t ^p 
    +  
    C  (  1 +  X (t) _V^2 p +  
     P_N X (t) _V^2 p )  (I - P_N) X(t) ^p.

Choosing γ =  p - 1  2 p ∈ 
(  p - 2  2 p ,  12) in Lemma <ref> 
and also considering (<ref>), (<ref>) and (<ref>) assure

    [  e^N_t ^p ]
       ≤
    C ∫_0^ t  [  e^N_s ^p ]  
    
    
    
    +
    
    [
          ( 1 +   X (s) _V^2p +  
     P_N X (s) _V^2p )  (I - P_N) X(s) ^p
         ]   s
    
       ≤
    C ∫_0^ t  [  e^N_s ^p ]  
    
    
    +
    
    
      (   
         1 +   X (s) _L^4p(Ω, V) ^2p +  
     P_N X (s) _L^4p(Ω, V) ^2p ) 
    
    
    
     (I - P_N) X(s) _L^2p(Ω, H) ^p  s
    
       ≤
    C ∫_0^ t  [  e^N_s ^p ]    s
    +
    C 
    N^- p βsup_s ∈ [0, T] X (s) _L^2p( Ω, Ḣ^β) ^p
    
         ×∫_0^t
     (
    1
    +
      X_0 ^6p_L^12p(Ω, V )
    +
    s^-1/2 X_0 ^2p_L^4p(Ω, Ḣ^(p-1)/(2p) ))
     s
    
       ≤
    C ∫_0^ t  [  e^N_s ^p ]    s
    +
    C
    N^- p β(
     1 +  X_0 _L^  p (Ω, Ḣ^β )  +  X_0 _L^  3p (Ω, V ) ^3 
             +
             A^β-1/2_ℒ_2 )^p
     
         ×( 1 
    +
      X_0 ^6p_L^12p(Ω, V )
     +
      X_0 ^2p_L^4p(Ω, Ḣ^(p-1)/(2p) ) )
    
       ≤
    C ∫_0^ t  [  e^N_s ^p ]    s
    +
    C
    N^- p β( 1 
    +
      X_0 ^6_L^12p(Ω, V )
     +
      X_0 _L^  4 p (Ω, Ḣ^β∨ [ (p-1)/(2p) ]  ) ^2
     +
     A^β-1/2_ℒ_2  )^2p.

After the use of the Gronwall inequality, one can derive from (<ref>) the desired error bound.
□



§ SPATIO-TEMPORAL FULL DISCRETIZATION




This section is devoted to error analysis of a spatio-temporal full discretization, 
done by a time discretization of the spatially discretized problem (<ref>).
For M ∈ we construct a uniform mesh on [0, T] with τ = TM being the time stepsize,
and propose a spatio-temporal full discretization as,

    Y^M,N_t_m+1 =    E_N(τ) Y_t_m^M,N +  A_N^-1 ( I - E_N( τ )  ) F_N (Y_t_m^M,N)  /  1 + τ F_N(Y_t_m^M,N) 
    + ∫_t_m^t_m+1 E_N(t_m+1-s) P_N  W(s)

for m = 0, 1, ..., M - 1 and Y^M,N_0 = P_N X_0.
Equivalently, the full discretization (<ref>) can be written by
Y^M,N_0 = P_N X_0 and for m = 0, 1, ..., M - 1,

    Y^M,N_t_m+1 =    E_N(τ) Y_t_m^M,N + ∫_t_m^t_m+1 E_N(t_m+1-s)  F_N (Y_t_m^M,N)  /  1 + τ F_N(Y_t_m^M,N)  s
    
    + ∫_t_m^t_m+1 E_N(t_m+1-s) P_N  W(s).




Here we invoke a taming technique in <cit.> 
for ordinary SDEs, and construct a nonlinearity-tamed accelerated exponential Euler (AEE) scheme as (<ref>).
The so-called AEE scheme without taming is originally introduced in <cit.>,
to strongly approximate nearly linear parabolic SPDEs. Since the stochastic convolution is 
Gaussian distributed and diagonalizable on {  e_i }_i ∈,  the scheme is much easier to simulate than 
it appears at first sight (see comments in section <ref> for the implementation).


When the nonlinearity grows super-linearly,  one can in general not expect that  
the usual AEE schemes <cit.> converge strongly,
based on the observations from 

<cit.>.  
Also, we mention that analyzing the strong convergence rate is much more difficult than that in the finite dimensional SDE setting. 

When analyzing strong convergence rates, a crucial element is to derive uniform moment bounds for the 
spatio-temporal full discretization. To do this, a key ingredient lies on transforming 
the continuous time extension (<ref>) of
the full discretization (<ref>) into a random differential equation (<ref>),
with a random perturbation appearing in the cubic nonlinear term. So the forthcoming subsection
attempts to reveal that the V-norm of the solution to the perturbed differential equation can be controlled by a norm of the perturbation 
(see Lemma <ref>). Based on this finding, one can further use the structure of the taming scheme
and a certain bootstrap argument (Lemma <ref> and (<ref>)) 
to provide a priori moment bound of the approximation, 
as shown by Theorem <ref> in subsection <ref>.  

Reecovering the higher temporal convergence  rate of order almost 12 also heavily hinges on 
commutativity properties of the nonlinearity (Lemma <ref>) and 
the improved temporal Hölder regularity of the solution in negative Sobolev space 
(Lemma <ref>), as presented in
subsection <ref>. Armed with these useful results, the expected convergence rate 
is finally proved in subsection <ref>.







 §.§ Useful estimates for a perturbed differential equation


















In the first part, we restrict ourselves to the following perturbed differential equation in H^N, N ∈,

    {[ ∂ v^N /∂ t  = - A_N  v^N + P_N F ( v^N + z^N ),
           t ∈ (0, T],;                                                     v^N ( 0 ) = 0, ].

where F comes from Assumption <ref> and z^N , v^N   [0, T] → H^N. 
In what follows, we aim to show that the V-norm of the solution to the perturbed problem (<ref>) 
can be controlled by the 𝕃^9-norm of the perturbation z^N in the nonlinear mapping (see Lemma <ref>), 
which plays a crucial role in deducing uniform moment bounds for the full discretization (<ref>). 
To show this, we need further smoothing properties of the analytic semigroup E(t), t > 0 described as follows.






Let t > 0 and let P_N, E (t) be defined as in the above sections. Then it holds that

    P_N E ( t ) ψ_L^p( D )   ≤ t^ - p-2/4pψ,
      ∀ p ≥ 2,
    
    
      P_N E ( t ) ψ_V    ≤ (  t  2  ) ^ - 1/2ψ_ L^1( D ) ,
    
     P_N E ( t ) ψ_ L^3( D )    ≤ (  t  2  )^ - 5/24ψ_ L^4/3( D ) .




 Proof of Lemma <ref>.   




Recalling (<ref>) with γ = 0 helps us to infer,

    P_N E ( t ) ψ_ L^p (D) ^p ≤ P_N E ( t ) ψ^2 · P_N E ( t ) ψ_ V ^p-2≤ψ^2 · t^ - p-2/4ψ^p-2 
    = 
    t^ - p-2/4ψ^p,


which validates the first assertion. To arrive at the second one, 
we again use (<ref>) with γ = 0 to get

    P_N E ( t ) ψ_V     =  P_N E ( t2 ) P_N E ( t2 )  ψ_V 
    
    ≤
    ( t2 )^ - 1/4 P_N E ( t2 )  ψ
    = 
    ( t2 )^ - 1/4sup_ϕ≤ 1  | ⟨ P_N E ( t2 )  ψ,  ϕ⟩ |
    
       
    =
    ( t2 )^ - 1/4sup_ϕ≤ 1  | ⟨ψ,  P_N E ( t2 ) ϕ⟩ |
    ≤
    ( t2 )^ - 1/4sup_ϕ≤ 1 ψ_ L^1(D) · P_N E ( t2 ) ϕ_V
    
       ≤
    ( t2 )^ - 1/2ψ_ L^1(D) .


Concerning the last inequality, one can similarly acquire

    P_N E ( t ) ψ_ L^3( D )    ≤ ( t2 )^ - 1/12 P_N E ( t2 )  ψ
    = 
    ( t2 )^ - 1/12sup_ϕ≤ 1  | ⟨ P_N E ( t2 )  ψ,  ϕ⟩ |
    
        =
    ( t2 )^ - 1/12sup_ϕ≤ 1  | ⟨ψ,  P_N E ( t2 ) ϕ⟩ |
    ≤
    ( t2 )^ - 1/12sup_ϕ≤ 1 ψ_ L^4/3 (D)  P_N E ( t2 ) ϕ_ L^4 ( D ) 
       ≤
    ( t2 )^ - 5/24ψ_ L^4/3 (D) .

The proof is now completed. □



It is easy to see, the perturbed problem (<ref>) has a unique solution in H^N, which can be expressed by

    v^N ( t ) = ∫_0^t E_N (t - s ) P_N F ( v^N (s) + z^N (s) )   s.

Define norms u _ 𝕃 ^q( D × [0, t]  )  : = 
 ( ∫_0^t  u (s) _L^q (D) ^q   s  )^1/q,   q ≥ 1,   t ∈ [ 0, T ]. For the particular case q = 2, 
𝕃 ^q( D × [0, t]  ) (𝕃 ^q for brevity) becomes a  Hilbert space with 
⟨ u, v ⟩_ 𝕃 ^2 ( D × [0, t]  )  : = ∫_0^t ⟨ u (s), v (s) ⟩ s.

The next lemma asserts that the V-norm of the solution to (<ref>) 
can be controlled by the 𝕃^9-norm of the perturbation z^N, which will be essentially used 
in proving moment bounds of the approximations.


 
Let v^N, N ∈ be the solution to (<ref>). For any t ∈ [ 0, T ], 
there exists a positive constant C, dependent of T but independent of N, such that

    v^N ( t ) _V 
    ≤ 
    C ( 1 +  z^N _𝕃^9 ( D × [0, t] ) ^9 ),
      ∀ t ∈ [0, T].


 Proof of Lemma <ref>. 
The assertion is trivial for t = 0. So we always suppose t > 0 in the following. 
We divide the proof into two steps. 



Step 1. For any fixed t ∈ ( 0, T], we claim first that, 
by setting ϱ_t :=  5 t^1/4max{ | a_2 |  | a_3 |  , |   a_1   a_3 |^1/2, 
 |   a_0   a_3  |^1/3},

    v^N _ 𝕃 ^4( D × [0, t] ) ≤ 
    5   z^N _ 𝕃 ^4( D × [0, t] )    or    v^N _ 𝕃 ^4( D × [0, t] ) ≤ϱ_t.

By deterministic calculus and noting  A_N v^N = A v^N for any v^N ∈ H^N,  we derive

    0 
       ≤12  v^N (t) ^2
    = 
     ∫_0^t ⟨ v^N ( s ),  
    - A_N v^N ( s ) 
    + 
    P_N F ( v^N(s) + z^N (s)  ) ⟩ s
    
       ≤∫_0^t ⟨ v^N ( s ),   
    F ( v^N(s) + z^N (s)  ) ⟩ s
    =
    ⟨ v^N, F ( v^N + z^N ) ⟩_ 𝕃 ^2 .


Noticing that a_3 < 0, for any v , z ∈,

    v f ( v + z )     = a_3 v ( v + z )^3 + a_2 v ( v + z )^2 + a_1 v ( v + z ) + a_0 v
    
       ≤
    a_3 v^4 + 3 a_3 v^3 z + a_3 v z^3 + a_2 v^3 + 2 a_2 v^2 z + a_2 v z^2 + a_1 v^2 + a_1 v z + a_0 v.

After using the fact a_3 < 0 and the Hölder inequality, one derives

    ⟨ v^N, F ( v^N + z^N ) ⟩_ 𝕃 ^2    ≤
    a_3  v^N _ 𝕃 ^4 ^4 
    +
    3 | a_3 |  v^N _ 𝕃 ^4 ^3  z^N _ 𝕃 ^4 
    +
    | a_3 |  v^N _ 𝕃 ^4  z^N _ 𝕃 ^4 ^3
    
          + 
    | a_2 | t^1/4 v^N _ 𝕃 ^4 ^3
    + 
    2 | a_2 | t^1/4 v^N _ 𝕃 ^4 ^2  z^N _ 𝕃 ^4 
    +
    | a_2 |  t^1/4 v^N _ 𝕃 ^4  z^N _ 𝕃 ^4 ^2
    
         
    + 
    | a_1 |  t^1/2 v^N _ 𝕃 ^4 ^2
    +
    | a_1 |  t^1/2 v^N _ 𝕃 ^4  z^N _ 𝕃 ^4 
    +
    | a_0 | t^3/4 v^N _ 𝕃 ^4  .

Assume the claim (<ref>) is false, namely, 
z^N _ 𝕃 ^4( D × [0, t] )  <  1 / 5  v^N _ 𝕃 ^4( D × [0, t] )
and
v^N _ 𝕃 ^4( D × [0, t] )    >  ϱ_t.
This enables us to derive, in the case ϱ_t > 0, i.e., | a_0 | + | a_1 | + | a_2 | > 0,

    ⟨ v^N, F ( v^N + z^N ) ⟩_ 𝕃 ^2     <
     ( a_3 +  3 | a_3 |  5  +  | a_3 |  125 )  v^N _ 𝕃 ^4  ^4
    +
    ( | a_2 | t^1/4 +   2 | a_2 |  t^1/4 5  +   | a_2 |  t^1/4 25 )  v^N _ 𝕃 ^4  ^3
    
          + 
     ( |a_1| t^1/2 +   | a_1 |  t^1/2 5 )  v^N _ 𝕃 ^4  ^2
    +
    | a_0 | t^3/4 v^N _ 𝕃 ^4 
        <
    ( 
    a_3 +  76 | a_3 |  125  
    +
     | a_2 | t^1/4ϱ_t  +   2 | a_2 |  t^1/4 5 ϱ_t  +   | a_2 |  t^1/4 25 ϱ_t  
    +
     |a_1| t^1/2ϱ_t^2  +   | a_1 |  t^1/2 5 ϱ_t^2 
    +
     | a_0 | t^3/4ϱ_t^3 ) 
     v^N _ 𝕃 ^4  ^4
    
       ≤( 
    a_3 +  76 | a_3 |  125  
    +
     | a_3 |  5  +   2 | a_3 |  25  +   | a_3 |    125  
    +
     | a_3 |  25  +   | a_3 |  125 
    +
     | a_3 |  125 ) 
     v^N _ 𝕃 ^4  ^4
    
        =
     6 a_3  125  v^N _ 𝕃 ^4  ^4
    < 0,

where we set ϱ_t :=  5 t^1/4max{ | a_2 |  | a_3 |  , |   a_1   a_3 |^1/2, 
 |   a_0   a_3  |^1/3} such that all coefficients of  v^N _ 𝕃 ^4  ^4 in the second step 
only consist of |a_3| or a_3 and the resulting collection turns out to be negative.
 
The conclusion (<ref>) thus contradicts (<ref>). 
When a_0  = a_1  = a_2 = 0, i.e., ϱ_t = 0,  one can similarly derive
⟨ v^N, F ( v^N + z^N ) ⟩_ 𝕃 ^2  
<
 ( a_3 +  3 | a_3 |  5  +  | a_3 |  125 )  v^N _ 𝕃 ^4  ^4
=
 49 a_3  125  v^N _ 𝕃 ^4  ^4
≤ 0,
also contradicting (<ref>).
Therefore, the claim (<ref>) must be true.


Step 2.
Apparently, (<ref>) implies

    v^N _ 𝕃 ^4 (  D × [0, t]   ) ≤ 
    5   z^N _ 𝕃 ^4 ( D × [0, t] )  
    + ϱ_T,
      ∀ t ∈ [0, T].


This together with the last inequality in (<ref>), the property of the cubic nonlinearity 
and the Hölder inequality yields, for any t ∈ [ 0, T],

    v^N ( t ) _L^3 ( D )    ≤
    ∫_0^t  E (t - s ) P_N F ( v^N (s) + z^N (s) ) _L^3 ( D )   s
    
       ≤
    ∫_0^t (  t - s2 )^ - 5/24  F ( v^N (s) + z^N (s) ) _L^4/3 ( D )   s
    
       ≤
    C 
    ∫_0^t (  t - s2 )^ -  5 / 24 ( 1 +  v^N (s) _ L^4 (D) ^3 +  z^N (s) _ L^4 (D) ^3 )   s
    
       ≤
    C 
     (  ∫_0^t (  t - s2 )^ -  5 / 6   s  )^1/4 (  ∫_0^t   1 +  v^N (s) _ L^4 (D) ^4 +  z^N (s) _ L^4 (D) ^4    s  )^3/4
       ≤
    C  ( 1 +  z^N _𝕃^4 ( D × [0, t] ) ^3  ).





















































Likewise, by virtue of the second inequality in (<ref>) instead, one obtains, 
for any t ∈ [ 0, T],

    v^N ( t ) _V  
       ≤
    ∫_0^t  P_N E ( t-s) F ( v^N (s) + z^N (s) ) _V   s
    
       ≤
    C 
    
    ∫_0^t (  t - s2 )^-1/2 F ( v^N (s) + z^N (s) ) _L^1(D)  s
    
       ≤
    C 
    
    ∫_0^t ( t - s )^-1/2  ( 1 +  v^N (s) _L^3(D)^3 +  z^N (s) _L^3(D)^3 )    s
    
       ≤
    C  ( 1 +  z^N _𝕃^4 ( D × [0, t] ) ^9 +  ∫_0^t  ( t - s )^-1/2 z^N (s) _L^3(D)^3   s   )
    
       ≤ 
    C ( 1 +  z^N _𝕃^9 ( D × [0, t] ) ^9 ).


The proof of Lemma <ref> is thus finished. □



Before closing this subsection, we would like to add some comments on the proof of Lemma <ref>.

The key step is to show either 
v^N _ 𝕃 ^4( D × [0, t] ) ≤κ z^N _ 𝕃 ^4( D × [0, t] ) 
or
v^N _ 𝕃 ^4( D × [0, t] ) ≤ϱ_t, 
with some positive constants κ, ϱ_t properly chosen.
As shown above, its proof is based on the reduction to absurdity and 
we choose κ = 5 and ϱ_t :=  5 t^1/4max{ | a_2 |  | a_3 |  , |   a_1   a_3 |^1/2, 
 |   a_0   a_3  |^1/3} in (<ref>) to result in a contradiction. 
We mention that the choice of κ, ϱ_t is, however, not unique.




 §.§ A priori moment bounds of the approximations





The aim of this subsection is to prove a priori moment bounds for the fully discrete approximation (<ref>)
(or (<ref>) equivalently), 
which is essentially based on the estimate (<ref>) obtained in the previous subsection 
as well as a certain bootstrap argument.
First, we define 



    ⌊ t ⌋ : =  t_i = i τ,     for  t ∈ [t_i, t_i+1), 
       
    i  ∈{ 0, 1, ... ,M-1} ,

and introduce a continuous version of the fully discrete scheme (<ref>) as, 

    Y^M,N_t 
    =     
    E_N(t) Y_0^M,N 
    + 
    ∫_0^t E_N(t - s)  F_N(Y^M,N_⌊ s ⌋)    1 + τ F_N(Y^M,N_⌊ s ⌋)  s 
    + 
    𝒪_t^N, 
    
      
    
     
    t ∈ [0, T].







Here we recall that τ = TM is the uniform time stepsize. 
By B^c and _B, we denote the complement and indicator function of a set B, respectively.
Additionally, we introduce a sequence of decreasing subevents

    Ω_R,t_i:= 
      {ω∈Ω: sup_j  ∈{ 0, 1, ... ,i} Y^M,N_ t_j (ω) _V ≤ R },
        
      R ∈ (0, ∞),   i  ∈{ 0, 1, ... ,M}.

It is clear that _Ω_R,t_i∈_t_i for i  ∈{ 0, 1, ... ,M} and 
_Ω_R,t_i≤_Ω_R, t_j
for t_i ≥ t_j since Ω_R,t_i⊂Ω_R, t_j, t_i ≥ t_j.

Furthermore, we put additional assumptions on the initial data.

For sufficiently large positive number p_0 ∈, the initial data X_0 obeys

    sup_ N ∈ P_N X_0 _L^p_0( Ω, V) < ∞.



Thanks to the Sobolev embedding inequality, (<ref>) is fulfilled provided P_N X_0 _L^p_0( Ω, Ḣ^γ) < ∞ for any γ > 12. In what follows we start the bootstrap argument, by showing
[ _Ω_ R^τ,t_m Y^M,N_t_m_V^p] < ∞ and [
   _Ω^c_R^τ,t_mY_t_m^M,N_V^p
    ] < ∞
for  subevents Ω_ R^τ,t  with R^τ depending on τ carefully chosen.


Let p ∈ [2, ∞) and R^τ := τ^-β/ 4 for any β∈ (0, 12). 
Under Assumptions <ref>-<ref>, <ref>, 

the approximation process Y^M,N_t_i, i ∈{ 0, 1,..., M }  produced by (<ref>) obeys


    sup_ M, N ∈sup_ i ∈{ 0, 1,..., M }[ _Ω_ R^τ, t_i-1 Y^M,N_t_i_V^p]
    <
    ∞,


where we set _Ω_R^τ,t_-1 = 1.


Proof of Lemma <ref>. 
 
 The proof heavily relies on the use of Lemma <ref>.
In order to apply it, we introduce a process Z_t^ M, N given by,


    Z_t^ M, N    :=  
     E_N(t) Y_0^M,N 
     +
       ∫_0^t E_N ( t - s )
       [
        P_N F( Y^M,N_⌊ s ⌋ )  1 + τ P_N F( Y^M,N_⌊ s ⌋ ) 
       -
       P_N F( Y^M,N_s )
       ]
         s
     +
     𝒪_t^N
     
        = 
     E_N(t) Y_0^M,N
     +
       ∫_0^t E(t-s) P_N
        [
       F(Y^M,N_⌊ s ⌋)
       -
       F(Y^M,N_s)
        ]
         s
    
    
    
          +
       ∫_0^t E(t-s)
        [
        P_N F ( Y^M,N_⌊ s ⌋ ) 1 + τ P_N F(Y^M,N_⌊ s ⌋) 
       -
       P_N F ( Y^M,N_⌊ s ⌋ )
        ]
         s
       +
       P_N 𝒪_t,
       
     t ∈ [0, T].

Bearing this in mind, one can rewrite (<ref>) as

    Y^M,N_t
        =
    
    
       ∫_0^t E_N (t - s) P_N F ( Y^M,N_s )   s
       +
       Z_t^ M, N.


Further, we define Y̅^M,N_t as

    Y̅^M,N_t
       :=
       Y^M,N_t - Z^M,N_t,
          with   Y̅^M,N_0
       =
       0
       .

So, we recast (<ref>) as

    Y̅^M,N_t
       =
    
    
       ∫_0^t E_N (t-s) P_N F ( Y̅^M,N_s+Z^M,N_s )
         s,
         
       t ∈ [0, T],

which satisfies

    / tY̅^M,N_t
       =
       - A_N Y̅^M,N_t
       +
       P_N F(Y̅^M,N_t+Z^M,N_t),
         
       t ∈ ( 0, T],
       Y̅^M,N_0
       =
       0.


Now one can employ Lemma <ref> to obtain, 

    Y̅^M,N_t _V ≤ C ( 1 +  Z^M,N_𝕃^9 ( D × [0, t] ) ^9 ),
      
    t ∈ [0, T],


where Z^M,N_· is defined by (<ref>).





































Thus, for any i ∈{ 0, 1,...,M },

    [ _Ω_R,t_i-1Y̅^ M, N _t_i_V^p ]
       ≤ 
    C 
    ( 1 + [ _Ω_R,t_i-1 Z^M,N_𝕃^9p( D × [0, t_i] )  ^9p] 
    )
    
       ≤ 
    C 
    ( 1 + [ 
    _Ω_R,t_i-1∫_0^t_i Z^M,N_s _V^9p  s ] 
    ),

where, for s ∈ [0, t_i], i ∈{ 0, 1,...,M }, it holds that

    _Ω_R,t_i-1 Z^M,N_s _V
       ≤
        E_N( s ) Y_0^M,N_V
       +
       _Ω_R,t_i-1∫_0^s 
       E ( s - r ) P_N
        [
       F( Y^M,N_r )
       -
       F( Y^ M,N _⌊ r ⌋ )
        ]
         r
       _V
    
    
    
    
    
    
    
          +
     _Ω_R,t_i-1∫_0^s E(s-r) P_N  F(Y^M,N_⌊ r ⌋) 
       τP_N F(Y^M,N_⌊ r ⌋)1 + τ P_N F(Y^M,N_⌊ r ⌋)   r
       _V
    
    
    
     +
        P_N 𝒪_s  _V
    
        =:
        E_N( s ) Y_0^M,N_V
       +
       I_1 + I_2 
       + 
        P_N 𝒪_s  _V.

Before proceeding further, we claim


    _Ω_R,t_i-1 Y_r^M,N_V 
    ≤ 
    C ( 
    1
    +
    R 
    + 
    τ^3/4 R^3 
    + 
    𝒪_r^N 
    _V
    +
    𝒪_⌊ r ⌋^N
    _V 
    ),
      ∀  r ∈ [0, t_i).


By the definition of Y_r^M,N and noting ∫_⌊ r ⌋^r E ( r - u ) P_N   W_u 
=  𝒪_r^N - E ( r - ⌊ r ⌋ )  𝒪_⌊ r ⌋^N,  r ∈ [0, t_i), one can write

    Y_r^M,N 
    =
    E ( r - ⌊ r ⌋ ) Y_⌊ r ⌋^M,N 
    +
    ∫_⌊ r ⌋^r   E ( r - u ) F_N ( Y_⌊ u ⌋^M,N )  1 + τ F_N ( Y_⌊ u ⌋^M,N )   u
    +
    𝒪_r^N - E ( r - ⌊ r ⌋ )  𝒪_⌊ r ⌋^N.

This together with the boundedness of the semigroup E (t) in V, i.e.,
E (t) ϕ_V ≤ϕ_V and (<ref>) for γ = 0 

promises

    _Ω_R,t_i-1 Y_r^M,N_V
       ≤_Ω_R,t_i-1(
     E ( r - ⌊ r ⌋ ) Y_⌊ r ⌋^M,N_V
    +
    ∫_⌊ r ⌋^r  E ( r - u ) F_N ( Y_⌊ u ⌋^M,N ) _V   u
    
         
    +
    𝒪_r^N - E ( r - ⌊ r ⌋ )  𝒪_⌊ r ⌋^N
    _V
    )
    
       ≤_Ω_R,t_i-1(
     Y_⌊ r ⌋^M,N_V
    +
    ∫_⌊ r ⌋^r ( r - u )^-1/4  F ( Y_⌊ u ⌋^M,N )   u
    +
    𝒪_r^N 
    _V
    +
    𝒪_⌊ r ⌋^N
    _V
    )
    
       ≤
    C 
    ( 
    1
    +
    R 
    + 
    43 τ^3/4 R^3 
    + 
    𝒪_r^N 
    _V
    +
    𝒪_⌊ r ⌋^N
    _V 
    )
    .





This shows the claim (<ref>).

With the aid of (<ref>) and (<ref>),  the first term I_1 can be treated as follows,

    I_1
        ≤_Ω_R,t_i-1∫_0^s (s-r)^- 1/4 F(Y^M,N_r)
       -
       F(Y^M,N_⌊ r ⌋)   r
    
       ≤_Ω_R,t_i-1 C 
       ∫_0^s 
       (s-r)^- 1/4 
       ( 1 + R^2 + τ^3/2 R^6 + 𝒪_r^N 
    _V^2
    +
    𝒪_⌊ r ⌋^N
    _V^2 )
        Y^M,N_r - Y^M,N_⌊ r ⌋  r,

where r ∈ [0, s], s ∈ [0, t_i],

    Y^M,N_r - Y^M,N_⌊ r ⌋    =
       [ E ( r ) - E ( ⌊ r ⌋ ) ] Y^ M, N_0 
       +
       ∫_0^r E(r-u)  P_N F (Y^M,N_⌊ u ⌋)  1 + τ P_N F (Y^M,N_⌊ u ⌋)    u
    
          
       -
       ∫_0^⌊ r ⌋ E(⌊ r ⌋-u)  P_N F (Y^M,N_⌊ u ⌋)  1 + τ P_N F (Y^M,N_⌊ u ⌋)    u
       + 
       P_N 𝒪_r
       -
       P_N 𝒪_⌊ r ⌋.














This ensures that


    _Ω_R,t_i-1 Y^M,N_r - Y^M,N_⌊ r ⌋
         ≤τ^β/2 Y^M,N_0 _β 
       + 
       1_Ω_R,t_i-1∫_0^⌊ r ⌋ E(⌊ r ⌋-u)
       (
       E( r - ⌊ r ⌋)-I )  P_N F (Y^M,N_⌊ u ⌋)  1 + τ P_N F (Y^M,N_⌊ u ⌋)    u
       
            +
       _Ω_R,t_i-1∫_⌊ r ⌋^r E ( r - u )  P_N F (Y^M,N_⌊ u ⌋)  1 + τ P_N F (Y^M,N_⌊ u ⌋)    u
       
    
       +
       _Ω_R,t_i-1 P_N ( 𝒪_r - 𝒪_⌊ r ⌋ ) 
       ≤τ^β/ 2  X_0 _β 
        + 
        C(1 + R^  3  ) (τ^3/4 + τ)
    
    
     +
       𝒪_r - 𝒪_⌊ r ⌋ .

Inserting this into (<ref>) results in

    I_1
        ≤ C ∫_0^s 
       ( 1 + R^2 + τ^3/2 R^6 + 𝒪_r^N 
    _V^2
    +
    𝒪_⌊ r ⌋^N
    _V^2 
    ) 
       ( s - r )^- 1/4 [
       τ^β/2 X_0 _β
       + C( 1+R^ 3  ) τ^3/4
       + 𝒪_r - 𝒪_⌊ r ⌋]
         r
    
       
       = C τ^β/2 X_0 _β∫_0^s 
       ( 1 + R^2 + τ^3/2 R^6 + 𝒪_r^N 
    _V^2
    +
    𝒪_⌊ r ⌋^N
    _V^2 
    )
    ( s - r )^ -1/4  r 
     
           
     +
       C (1 + R^3  ) τ^3/4∫_0^s
       ( 1 + R^ 2  + τ^3/2 R^6 + 𝒪_r^N 
    _V^2
    +
    𝒪_⌊ r ⌋^N
    _V^2  
    ) ( s - r )^ -1/4  r 
    
          
     +
       C ∫_0^s ( s - r)^- 1/4 ( 1 + R^2 + τ^3/2 R^6 
       +
       𝒪_r^N 
    _V^2
    +
    𝒪_⌊ r ⌋^N
    _V^2 
    ) 𝒪_r - 𝒪_⌊ r ⌋  r.


Therefore, letting R = R^τ := τ^-β/ 4 
and considering (<ref>)-(<ref>) 
one can further infer that

    I_1 _L^9p (Ω, ) ≤ 
    C 
    ( 
    1 
    + 
     X_0 _L^9p (Ω, Ḣ^β ) 
    +
    A^β-1/2_ℒ_2 (H)  ).






























In a similar manner, choosing R = R^τ := τ^-β/ 4 enables us to treat I_2 as follows:

    I_2
        ≤_Ω_R,t_i-1∫_0^s  E( s -r ) P_N F(Y^M,N_⌊ r ⌋) _V ·τ P_N F(Y^M,N_⌊ r ⌋)  r
    
    
    
    
    
    
    
    
       ≤_Ω_R,t_i-1τ∫_0^s ( s - r )^ - 1 / 4  F(Y^M,N_⌊ r ⌋) ^2
         r
    
       ≤ C (R^6 + 1) τ
       ≤ C ( τ^ 2 - 3β/2 + τ ),

where we also used (<ref>) with γ = 0 

and the structure of the taming scheme, namely,
z  1 + τ z ≤ z,  z ≥ 0.










Bearing (<ref>), (<ref>) and (<ref>) in mind, 
one can deduce from (<ref>) that, for any s ∈ [0, t_i],

    [ _Ω_R,t_i-1 Z^M,N_s _V ^ 9p ] ≤ C < ∞.

This together with (<ref>) and (<ref>) immediately implies

    [ _Ω_R^τ,t_i-1Y̅^M,N_t_i_V^p] ≤ C < ∞ .

Combining this with (<ref>) verifies the desired assertion (<ref>).
□


As a direct consequence of Lemma <ref>, the following corollary
offers a priori moment bounds of the numerical approximation on subevents Ω_ R^τ,t_i, i ∈{ 0, 1,..., M }.



Let p ∈ [2, ∞) and R^τ := τ^-β/ 4 for any β∈ (0, 12). 
Let Assumptions <ref>-<ref>, <ref> hiold

and let the approximation process Y^M,N_t_i, i ∈{ 0, 1,..., M }  be produced by (<ref>).
Then it holds

    sup_ M, N ∈sup_ i ∈{ 0, 1,..., M }[ _Ω_ R^τ, t_i Y^M,N_t_i_V^p]
    <
    ∞.



Proof of Corollary <ref>. 
Thanks to the assertion (<ref>) and the fact that Ω_R,t_i⊂Ω_R,t_i-1, one can easily deduce

    sup_ M, N ∈sup_ i ∈{ 0, 1,..., M }[ _Ω_ R^τ, t_i Y^M,N_t_i_V^p]
    ≤sup_ M, N ∈sup_ i ∈{ 0, 1,..., M }[ _Ω_ R^τ, t_i-1 Y^M,N_t_i_V^p]
    <
    ∞,

as required. □


Equipped with the bounded moments (<ref>) on subevents Ω_ R^τ,t_i, i ∈{ 0, 1,..., M }, 
one only needs to deduce bounded moments on its complement Ω_R^τ,t_m^c
before arriving at the expected bounded moments on the whole probability space stated as follows.


Let Assumptions <ref>-<ref>, <ref> be fulfilled
and let the approximation process Y^M,N_t_i, i ∈{ 0, 1,..., M }  be produced by (<ref>). 

Then for any p ∈ [2, ∞) it holds that


    sup_ M, N ∈sup_ m ∈{ 0, 1,..., M }[  Y^M,N_t_m_V^p]
    <
    ∞.


Proof of Theorem <ref>.
As discussed above, it remains to bound
sup_ M, N ∈sup_ m ∈{ 0, 1,..., M } [ _Ω_R^τ,t_m^c Y^M,N_t_m_V^p].
Owing to the boundedness of the semigroup E (t) in V, i.e.,
E (t) ϕ_V ≤ϕ_V, (<ref>) with γ = 0


and the structure of the taming scheme, i.e. z  1 + τ z ≤τ^-1 for z ≥ 0, 
one can infer 


    Y_t_m^M,N_V
        ≤E( t_m ) P_N X_0 _V
       +
        P_N 𝒪_t_m_V
       +
       ∫_0^t_m
       E( t_m - s ) P_N   F(Y^M,N_⌊ s ⌋)1+τ P_N   F(Y^M,N_⌊ s ⌋) _V
         s
    
    
    
    
    
    
    
    
    
    
    
    
       ≤ P_N X_0 _V
       + 
       ∫_0^t_m
       (t_m - s)^-1/4 P_N
         F(Y^M,N_⌊ s ⌋) 1 + τ P_N F(Y^M,N_⌊ s ⌋)   s
       +  P_N 𝒪_t_m_V
       
       ≤
        P_N X_0 _V
       +  43 t_m^3/4τ^-1
       + 
        P_N 𝒪_t_m_V,
         
       m ∈{ 0, 1, 2,..., M }.


Thanks to Lemma <ref> and Assumption <ref>, we can show that for any p ≥ 2,

    Y_t_m^M,N_L^p( Ω, V)≤
    C ( 1 + τ^-1 ),
      
       m ∈{ 0, 1, 2,..., M }.











Meanwhile, one can learn that

    Ω^c_R^τ,t_m
       =
       Ω^c_R^τ,t_m-1
       +
       Ω_R^τ,t_m-1·{ω∈Ω :  Y^M,N_t_m_V > R^τ},

and as a result

    _Ω^c_R^τ,t_m    =
       _Ω^c_R^τ,t_m-1
       +
       _Ω_R^τ,t_m - 1·_{ Y^M,N_t_m_V > R^τ}
    
    =
       ∑_i=0^m _Ω_R^τ, t_i-1·_{ Y^M,N_t_i_V > R^τ} ,

where we recall
_Ω^c_R^τ,t_-1 = 0.
By (<ref>) and the Chebyshev inequality, one can show, 
for any m ∈{ 0, 1,...,M }, M ∈,

    [
       _Ω^c_R^τ,t_mY_t_m^M,N_V^p
        ]
    
        
     =
       ∑_i=0^m  [
       Y_t_m^M,N_V^p ·_Ω_ R^τ,t_i-1_{ Y^M,N_t_i_V > R^τ} ]
     
       ≤∑_i=0^m
        (  [
       Y_t_m^M,N_V^2p ]
        )^1/2(
        [
       _Ω_R^τ, t_i-1_{ Y^M,N_t_i_V > R^τ} ]
        )^1/2
       ≤∑_i=0^m C ( 1+τ^-p)
       (
       ℙ(
       ω∈Ω_R^τ,t_i-1 :  Y^M,N_t_i_V
       > R^τ)
       )
       ^1/2
       
       = 
       C ( 1+τ^-p)
       ∑_i=0^m
       (
       ℙ(
       ω∈Ω :
       _Ω_R^τ,t_i-1 Y^M,N_t_i_V
       > R^τ)
       )
       ^1/2
       ≤ C ( 1+τ^-p)
    
       ∑_i=0^m
        (
        [
       _Ω_R^τ, t_i-1 Y^M,N_t_i_V^8(p+1)/β / (R^τ)^8(p+1)/β ]
        )
       ^1/2
       ≤
        C ( 1+τ^-p)
    
       ∑_i=0^m
       τ^ p+1  (
        [
       _Ω_R^τ, t_i-1 Y^M,N_t_i_V^8(p+1)/β ]
        )
       ^1/2
    
       < ∞.

















This estimate together with (<ref>) yields
the required estimate (<ref>).
□

With Theorem <ref> at hand, one can use standard arguments to obtain the coming corollaries.













 
Let Assumptions <ref>-<ref>, <ref> be fulfilled
and let the approximation process Y^M,N_t, t ∈ [0, T] be given by (<ref>).
Then for any p ∈ [2, ∞) and β < 12 we have

    sup_ M, N ∈,   t ∈ [0, T] 
    Y^M,N_t
    _ L^p ( Ω, Ḣ^β ) 
    +
    sup_ M, N ∈,   t ∈ [0, T] 
    Y^M,N_t
    _ L^p ( Ω, V ) 
    <
    ∞.




Let Assumptions <ref>-<ref>, <ref> be fulfilled
and let the approximation process Y^M,N_t, t ∈ [0, T] be given by (<ref>). 
Then for any p ∈ [2, ∞) and β < 12 there exists a constant C, 
depending on p, β, T and the initial data but not depending on M,N, such that

    Y_t^M,N - Y^M,N_s _ L^p ( Ω, H ) ≤ C (t-s)^β/2 , 
           
         0 ≤ s<t ≤ T.







 §.§ Refined temporal Hölder regularity results in negative Sobolev spaces



In addition to the a priori moment bounds for the approximations, we further rely on 
refined temporal Hölder regularity results in negative Sobolev spaces, as stated in Corollary  <ref>,
which are essentially used in proving the temporal convergence rates of order almost 12.
To arrive at Corollary <ref>, we also need commutativity properties of the nonlinearity 
 and the improved temporal Hölder regularity of the solution in negative Sobolev spaces, 
as presented in Lemma <ref> and Lemma <ref>, respectively.

For the first step, we reveal commutativity properties of the nonlinearity in the forthcoming lemma.




Let 

F  L^6 ( D; ) → H 
be a mapping determined by Assumption <ref>.
Then for any β∈ (0, 12) and η > 12 there exists a constant C
depending on β, η, {a_i}_i=0^3 such that

    F'( χ ) ν_ - η≤
    C (1+max{χ_V ,χ_β}
       ^2)  ν_-β,
      ∀χ∈ V ∩Ḣ^β, ν∈ V.




Proof of Lemma <ref>. 
As β∈ (0, 1/2 ), standard arguments with the Sobolev-Slobodeckij norm (see, e.g., <cit.>) 
and properties of the nonlinear mapping guarantee

    F' ( χ ) ψ_β^2
        ≤
       C  F' ( χ ) ψ^2
       +
       C ∫_0^1 ∫_0^1 |f'(χ(x)) ψ(x)
       -
       f'(χ(y)) ψ(y)|^2/|x-y|^2β+1  y  x
    
       ≤
       C  F ' ( χ ) ψ^2
       +
       C ∫_0^1 ∫_0^1 |f'(χ(x)) (ψ(x)-ψ(y))|^2/|x-y|^2β+1  y  x
    
          +
       C ∫_0^1 ∫_0^1
       | [ f'(χ(x))- f'(χ(y)) ] ψ(y)|^2/|x-y|^2β+1  y  x
    
       ≤
       C  F ' ( χ ) ψ^2
       +
       C   f'(χ(·)) _V^2 ·ψ^2_W^β,2
    
    
     +
       C
       
       f”( χ(·) )
       _V^2
       ·ψ_V^2
       ·χ^2_W^β,2
       ≤
       C ( 1+ χ_V^ 4 )  ψ^2
       +
       C  ( 1+ χ_V^ 4 ) ψ^2_β
    
    
     +
       C ( 1 +  χ_V^ 2 )
       ψ^2_V ·χ^2_β
       ≤
       C  (1+max{χ_V ,χ_β}
       ^4)
       (ψ^2_β + ψ^2_V ).

Accordingly, for any β∈ (0, 12) and η > 12, one uses the Sobolev embedding inequality to derive

    F' ( χ ) ν_ - η   
       =
       sup_φ≤ 1 |⟨ A^-η/2
       F' ( χ ) ν ,φ⟩|
    
    
       =
       sup_φ≤ 1 |⟨ν,
       ( F' ( χ ) )^∗· A^-η/2φ⟩|
    
       
       =sup_φ≤ 1 |⟨ A^-β/2ν , A^β/2 F' ( χ )
        A^-η/2φ⟩|
    
       ≤sup_φ≤ 1 ν_-β· F' ( χ ) A^-η/2φ_β
       ≤sup_φ≤ 1 ν_-β·
       C (1+max{χ_V ,χ_β}
       ^ 2 )
    
    
    
       (φ_β-η+  A^-η/2φ_V )
    
       ≤
       C_β (1+max{χ_V ,χ_β}
       ^ 2 ) ν_-β.

This completes the proof.
□


For the second step, we establish the improved temporal Hölder regularity of the solution in negative Sobolev spaces
as follows.

 


Let Assumptions <ref>-<ref>, <ref> be fulfilled
and let the approximation process Y^M,N_t, t ∈ [0, T]  be produced by (<ref>).
Then for any p ∈ [2, ∞), δ∈ [0, 12] and β < 12 
there exists a constant C depending on T, p, δ, β, {a_i}_i=0^3 
and the initial data but independent of M,N such that

    Y_t^M,N - Y^M,N_s _ L^p ( Ω, Ḣ^ -δ ) ≤ C (t-s)^β + δ/2 , 
           
         0 ≤ s < t ≤ T.



Proof of Lemma <ref>. 
The definition (<ref>) implies, for 0 ≤ s < t ≤ T,

    Y_t^M,N-Y^M,N_s
        
       =
       (
       E_N ( t - s ) - I
       )
       Y^M,N_s
        
          
          +
       ∫_s^t E_N ( t - r ) 
        F_N(Y^M,N_⌊ r ⌋)1+τ
       F_N(Y^M,N_⌊ r ⌋)
         r
       +
       ∫_s^t E_N ( t - r ) P_N   W ( r ).












Making use of (<ref>), (<ref>) and the inequality 
ΓΓ_1_ℒ_2 ≤Γ_ℒΓ_1_ℒ_2, 
Γ∈ℒ(H), Γ_1 ∈ℒ_2(H) gives

    ∫_s^t E_N ( t - r ) P_N   W ( r ) 
     _ L^p ( Ω, Ḣ^- δ )    ≤
       C
        (
       ∫_s^t  A^-δ/2
       E_N ( t - r ) P_N
       ^2_ℒ_2(H)   r
       )^ 1 /2
    
       ≤
       C
       (t-s)^β + δ/2.

A combination of (<ref>) and Corollary <ref> shows

    (
       E_N ( t - s ) - I
       )
       Y^M,N_s
    _ L^p ( Ω, Ḣ^-δ )    ≤
       A^- β + δ/2(
       E ( t - s ) - I
       ) _ℒ (H) ·
       Y^M,N_s
       _ L^p ( Ω, Ḣ^β ) 
       ≤
       C (t-s)^β + δ/2.


Now we  proceed to estimate the remaining term in (<ref>), 
with the help of (<ref>) and (<ref>),


    ∫_s^t E(t - r)  
        F_N(Y^M,N_⌊ r ⌋)1+τ
       F_N(Y^M,N_⌊ r ⌋)
         r
     _ L^p ( Ω, Ḣ^-δ ) ≤ 
       C
       ∫_s^t 
        F (Y^M,N_⌊ r ⌋)
       _ L^p ( Ω, H )   r
       ≤
       C ( t - s ) .
 
Gathering (<ref>), (<ref>) and (<ref>) 
we deduce from (<ref>) that (<ref>) is true.

□



It is worthwhile to mention that  the approximate solution (<ref>) enjoys higher order 
temporal Hölder regularity in negative Sobolev spaces Ḣ^δ, δ >0,
as indicated by Lemma <ref>. 
More precisely, the order of temporal Hölder regularity is increased to be β + δ2 when measured 
in the negative Sobolev space Ḣ^δ, δ∈ (0, 12], in contrast to order β2 in H by 
Corollary <ref>. So taking δ = 0 in (<ref>)
reduces into the usual temporal Hölder regularity results in H, i.e., (<ref>)
in Corollary <ref>.




Combining Lemmas <ref>, <ref> with Corollary <ref>,
one can easily arrive at the following corollary, which is used several times in the error analysis,
to improve the strong convergence rate of the scheme.
 

Let Assumptions <ref>-<ref>, <ref> be fulfilled
and let the approximation process Y^M,N_t, t ∈ [0, T]  be produced by (<ref>).
Then for any p ∈ [2, ∞), β < 12 and η > 12,
there exists a constant C depending on T, p, η, β, {a_i}_i=0^3 
and the initial data but independent of M,N such that

    F ( Y_t^M,N ) - F ( Y_s^M,N ) _ L^p ( Ω, Ḣ^ - η ) ≤
    C ( t - s )^β, 
           
         0<s<t<T.






 §.§ Main results: error bounds for the full discretization




At the moment, we are well prepared to prove the expected strong convergence rate of the proposed scheme, 
as stated in the following theorem.






Let Assumptions <ref>-<ref>, <ref> hold. 
Let X (t) and Y^M,N_t, t ∈ [0, T] be defined by 
(<ref>) and (<ref>), respectively.


For any p ∈ [2, ∞) and β∈ (0, 12), 
there exists a constant C depending on T, p, β, {a_i}_i=0^3 
and the initial data but independent of M,N such that

    sup_ t ∈ [0, T] X( t ) - Y^M,N_t_L^p ( Ω; H) ≤
    C (
         N ^ - β + τ ^β).





Proof of Theorem <ref>. 
Denoting e^ M, N_t : = P_N X(t) - Y_t^ M, N, we note that

    X( t ) - Y^M,N_ t _L^p ( Ω; H) ≤
     ( I - P_N ) X( t ) _L^p ( Ω; H) 
    +
    
     e^M,N_t_L^p ( Ω; H) ,



where 

    / t e^ M, N_t
    
    
    
       =
       - A_N
       e^ M, N_t
       +
       F_N
        (
       X ( t )
        )
       -
        F_N(Y^M,N_⌊ t ⌋)1+τ
       F_N(Y^M,N_⌊ t ⌋)
        .

This in conjunction with (<ref>) tells us that

    e^ M, N_t
       ^p
           =
       p ∫_0^t
       
       e^ M, N_s
       ^p-2
       ⟨
       e^ M, N_s, 
       - A_N
       e^ M, N_s
    
    
    
       +
       F_N
        (
       X (s) )
       -
        F_N(Y^M,N_⌊ s ⌋) 1 + τ
        F_N ( Y^M,N_⌊ s ⌋)
        ⟩  s
    
       ≤  
    p ∫_0^t
       
       e^ M, N_s
       ^p-2
       ⟨
       e^ M, N_s, 
       F_N ( X (s) ) - F_N ( P_N X (s) ) 
       
             
       +
       F_N
        (
       Y^M,N_s
        )
        -
        
       F_N
       (
       Y^M,N_⌊ s ⌋
       )
        1 + τ
       F_N(Y^M,N_⌊ s ⌋)
       ⟩  s
        +
        pL_0 ∫_0^t
        
        e^ M, N_s
        ^p
          s
    
         =
       p L_0 ∫_0^t
       
        e^ M, N_s
        ^p
          s
    
    
    
       +
       p ∫_0^t
       
       e^ M, N_s
       ^p-2
       ⟨
       e^ M, N_s, F( X (s) ) - F ( P_N X (s)  )
       ⟩  s
       
          +
       p ∫_0^t
       
       e^ M, N_s
       ^p-2
       ⟨
       e^ M, N_s, F(Y_s^M,N) - F (Y^M,N_⌊ s ⌋)
       ⟩  s
    
          +
       p ∫_0^t
       
       e^ M, N_s
       ^p-2⟨
       e^ M, N_s,
       τ
       F_N(Y^M,N_⌊ s ⌋)
       ·
       F(Y^M,N_⌊ s ⌋) 1+τ
       F_N(Y^M,N_⌊ s ⌋)
       ⟩  s
    
         =:
       p L_0 ∫_0^t
       
       e^ M, N_s
       ^p
         s
       +
       J_0
       + 
       J_1 
       + 
       J_2.


Following the same lines as in estimates of (<ref>) and (<ref>)
one can bound the item J_0 as follows,

    [ J_0 ] 
       ≤
    p  ∫_0^t
       
       e^ M, N_s
       ^p-1
       
       F( X (s) ) - F ( P_N X (s)  )
         s
    
       ≤
    ( p - 1 ) ∫_0^t
        [
       
       e^ M, N_s
       ^p
       ]
         s
    +
    ∫_0^t
     
     [
       
       F( X (s) ) - F ( P_N X (s)  )
       ^p
     ]
       s
    
       ≤
    ( p - 1 ) ∫_0^t
        [
       
       e^ M, N_s
       ^p
       ]
         s
    +
    C
    ( 1N )^ p β.


The term J_2 is also easy to be treated, after taking 

the Hölder inequality and (<ref>)
into account:

    [ J_2 ]
        ≤ p  ∫_0^t
       
       e^ M, N_s
       ^p-1·τ
       F(Y^M,N_⌊ s ⌋)
       ^2
         s
    
       ≤
       (p-1)
       ∫_0^t
       [
         
         e^ M, N_s
       ^p
       ]
         s
       +
       τ^p ∫_0^t
       [
       
       F(Y^M,N_⌊ s ⌋)
       ^2p]
         s
    
    
    
    
    
    
       ≤
       (p-1) ∫_0^t   [
       
       e^ M, N_s
       ^p
        ]
         s
       +
       C
    
    
    
    
    
    
    
    
       τ^p .

The remaining term J_1 must be handled more carefully.  
As usual, such a term is simply treated with the aid of temporal Hölder regularity of Y_s^M,N
together with the Cauchy-Schwarz inequality and Hölder's inequality,
but to only attain order τ^β/2. 
In our analysis, we decompose P_N X( s ) - Y_s^ M, N 
in the inner product into three parts.

To do so we recall that 

    e^ M, N_s = P_N X ( s ) - Y_s^ M, N  
    =
       ∫_0^s E(s-r)
        (
       F_N
        (
       X ( r )
        )
       -
        F_N(Y^M,N_⌊ r ⌋)1+τ
       F_N(Y^M,N_⌊ r ⌋)
        ) 
        r,

and split J_1 into three terms:


    J_1
         =
       p ∫_0^t
       
       e^ M, N_s
       ^p-2⟨∫_0^s E(s-r)
        (
       F_N
        (
       X ( r )
        )
       -
        F_N(Y^M,N_⌊ r ⌋)1+τ
       F_N(Y^M,N_⌊ r ⌋)
        )
         r,
    
    
    
       F(Y_s^M,N)-F(Y^M,N_⌊ s ⌋)
       ⟩  s
    
        =
       p ∫_0^t
       
       e^ M, N_s
       ^p-2⟨∫_0^s E(s-r)
        (
       F_N
        (
       X ( r )
        )
       -
       F_N(Y^M,N_ r  )
        )
        r ,
    
    
       F(Y_s^M,N)-F(Y^M,N_⌊ s ⌋)
       ⟩  s
    
          +
       p ∫_0^t
       
       e^ M, N_s
       ^p-2⟨∫_0^s E(s-r)
        (
       F_N(Y_r^M,N)-F_N(Y^M,N_⌊ r ⌋)
       )
         r,
    
    
       F(Y_s^M,N)-F(Y^M,N_⌊ s ⌋)
       ⟩  s
    
          +
       p ∫_0^t
       
       e^ M, N_s ^p-2⟨∫_0^s E(s-r) τ
       F_N(Y^M,N_⌊ r ⌋)
       ·
       F_N(Y^M,N_⌊ r ⌋) 1+τ
       F_N(Y^M,N_⌊ r ⌋)
         r ,
    
    
       F(Y_s^M,N)-F(Y^M,N_⌊ s ⌋)
       ⟩  s
    
        =:
       J_11 + J_12 + J_13.

Since the estimates concerning J_11 and J_12 are demanding, we handle the item J_13 first.
Utilizing (<ref>), the Hölder inequality, (<ref>) and  
(<ref>) results in

    [ | J_13 | ]
          ≤ p   ∫_0^t ∫_0^s
       
       e^ M, N_s
       ^p-2·τ
       F(Y^M,N_⌊ r ⌋)
       ^2
       · F(Y_s^M,N)-F(Y^M,N_⌊ s ⌋)
         r  s
    
       ≤
       C τ∫_0^t ∫_0^s
       
       e^ M, N_s
       ^p-2
        F(Y^M,N_⌊ r ⌋)
       ^2 
    
        (
       1
       +
       
       Y_s^M,N_V^ 2 
       +
       
       Y^M,N_⌊ s ⌋_V^ 2  )
    
    
    
        Y_s^M,N-Y^M,N_⌊ s ⌋  r  s
    
       ≤ C ∫_0^t  
       [
       
       e^ M, N_s
       ^p
       ]
         s
       +
       C τ^p/2∫_0^t 
       [
       |
       ∫_0^s
       
       F(Y^M,N_⌊ r ⌋)
       ^2
       · (
       1+ Y_s^M,N_V^2
          +
        Y^M,N_⌊ s ⌋_V^2 )
        Y_s^M,N-Y^M,N_⌊ s ⌋  r
        |^p/2]
         s
    
       ≤ C ∫_0^t 
       [
       
       e^ M, N_s
       ^p
       ]
         s
       +
       C
       τ^p/2(1+  β/2 ).

At the moment we come to the estimate of J_11 and use the Taylor formula, 
the self-adjointness of operators F ' ( u ) and P_N to infer that

    [ | J_11 | ]
           =
       p   [ |
       ∫_0^t
       
       e^ M, N_s
       ^p-2⟨∫_0^s E(s - r)
       (
       F_N ( X ( r ) ) - F_N ( Y_r^M,N )
        )
         r ,
    
    
    
       F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ⟩  s
        | ]
    
        =
       p   [ |
       ∫_0^t
       
       e^ M, N_s
       ^p-2⟨∫_0^s E(s - r )
       P_N ∫_0^1 F'
       ( Y_r^M,N + σ ( X ( r ) - Y_r^M,N )
       )
       σ
               ·(
       X ( r ) - Y_r^M,N)
         r ,
          F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ⟩  s
        | ]
     
        =
       p   [ |
       ∫_0^t ∫_0^s ∫_0^1
       
       e^ M, N_s
       ^p-2⟨
       X ( r ) - Y_r^M,N,
       (
       F' ( Y_r^M,N + σ ( X ( r ) - Y_r^M,N )
       )
       )^*
    
                   
        P_N E(s - r)
       [
       F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ]
       ⟩ σ  r   s
        | ]
    
       ≤
       p  ∫_0^t ∫_0^s ∫_0^1
       
       e^ M, N_s
       ^p-2·
       X ( r ) - Y_r^M,N·
        F'
       ( Y_r^M,N + σ ( X ( r ) - Y_r^M,N )
       )
    
                      
        P_N A^η/2 E(s - r) A^ - η/2[
       F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ]
        σ  r   s
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        ≤
      C  ∫_0^t ∫_0^s
       
       e^ M, N_s
       ^p-2
       X ( r ) - Y_r^M,N· ( 1 +  X ( r ) _V^2 +  Y_r^M,N_V^2
         )
    
                   
    
    × ( s - r )^- η/2A^-η/2[
       F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ]
         r   s.


Further, employing the Young inequalities a b ≤a^δδ +  b^δ'δ' for
a , b ≥ 0, δ, δ' > 1, 1δ +  1 δ' = 1,, the Hölder inequality,
the transformation of integral domain and taking 12 < η < 1 give

    [ | J_11 | ]
       ≤
       C  ∫_0^t ∫_0^s
    
        ( s - r )^ - η/ 2 
       e^ M, N_s
       ^p  r   s
       +
       C  ∫_0^t 
    
       ∫_0^s 
       ( s - r )^ - η/ 2 
       X ( r ) - Y_r^M,N^p/2
                   × ( 1 +  X ( r ) _V^p +  Y_r^M,N_V^p
         )
       A^-η/2[
       F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ]
       ^p/2  r 
    
         s
    
       ≤
       C ∫_0^t  [
         
       e^ M, N_s
         ^p ]
         s
       +
       C  ∫_0^t 
       ∫_0^s 
       ( s - r )^ - η/ 2 
       X ( r ) - Y_r^M,N^p  r   s
       
          +
       C  ∫_0^t 
       ∫_0^s 
       ( s - r )^ - η/ 2  ( 1 +  X ( r ) _V^2p +  Y_r^M,N_V^2p )
       A^-η/2[
       F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ]
       ^ p   r   s
    
       ≤
       C ∫_0^t  [
         
       e^ M, N_s
         ^p ]
         s
       +
       C ∫_0^t  [
         
       X ( s )-Y_s^M,N^p ]
         s
       
         
       +
       C ∫_0^t ∫_0^s 
       ( s - r )^ - η/ 2  [ 
        ( 1 +  X ( r ) _V^2p +  Y_r^M,N_V^2p )
       A^-η/2[
       F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ]
       ^p
        ]
         r   s,

where the first step is trivial for p = 2 due to (<ref>) and holds true for p > 2 
after using the aforementioned Young inequality with δ = p/p-2, δ' = p/2.


To proceed further,

we resort to Corollary <ref> as well as (<ref>),  
(<ref>) and achieve

    [ | J_11 | ]
       
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        ≤
       C  ∫_0^t  [
        e^ M, N_s ^p ]   s
    
    
       +
       C λ_N + 1^-  p β/2
       +
       C  ∫_0^t ( 
       [
        A^-η/2[
       F(Y_s^M,N)- F(Y^M,N_⌊ s⌋)
       ]
       ^2p]
       )^1/2  s
    
       ≤
       C ∫_0^t   [ 
        e^ M, N_s ^p  ]   s
       +
       C   ( 1N )^ p β 
       +
       C  τ^p β .



Finally, it remains to deal with the estimate of  J_12.
By the Young inequalities a b ≤a^δδ +  b^δ'δ' for
a , b ≥ 0, δ, δ' > 1, 1δ +  1 δ' = 1
and putting 12 < η < 1 one can derive that

    [ | J_12 | ]
          
       =
       p [ |
       ∫_0^t   
       
       e^ M, N_s  
        ^ p-2⟨∫_0^s  A^η E(s-r) A^ -η/2 ( F_N   ( Y^M,N_r   )
       -
       F_N   ( Y ^M,N  _ ⌊ r ⌋ )   )    r,
    
               
       A ^  -η/2 (  F_N   ( Y^M,N_s   )
       -
        F_N  ( Y^M,N_⌊ s ⌋)  ) ⟩  s
         | ]
    
       ≤
        C  ∫_0^t  ∫_0^s
        (s-r)^-η
        e^ M, N_s  
        ^p-2· A^-η/2(F  (  Y^M,N_r  )
         -
        F ( Y^M,N_⌊ r ⌋) )  
               ×  A^-η/2( F  (Y^M,N_s  )
        -
        F  ( Y^M,N_⌊ s ⌋) )   r   s
    
       ≤
       C ∫_0^t ∫_0^s
       (s-r)^-η  [ 
       e^ M, N_s
       ^p  ]    r   s
    
    
    
       +
       C  ∫_0^t  ∫_0^s (s-r)^-η A^-η/2( F  ( Y^M,N_r  )
       -
       F (Y^M,N_⌊ r ⌋))  ^p/2
                   × A^-η/2( F (Y^M,N_s  )
       -
       F ( Y^M,N_⌊ s ⌋) )  ^p/2  r   s
    
       ≤
      C ∫_0^t 
         [ 
       e^ M, N_s
       ^p  ]    s
    
       +
       C  ∫_0^t  ∫_0^s (s-r)^-η[   A^-η/2( F  ( Y^M,N_r  )
       -
       F (Y^M,N_⌊ r ⌋))  ^p ]
         r   s
     
         
       +
       C  ∫_0^t  ∫_0^s (s-r)^-η[   A^-η/2( F  ( Y^M,N_s  )
       -
       F (Y^M,N_⌊ s ⌋))  ^p ]
         r   s
       ,


where the last step but one is trivial for p = 2 and holds true for p > 2 after using the aforementioned 
Young inequality with δ = p/p-2, δ' = p/2. In the last step, we used the inequality
a b ≤a^22 + b^22, a, b ≥ 0. 
Again, the use of Corollary <ref> leads us to

    [ | J_12 | ] 
    ≤ 
    C ∫_0^t 
         [ 
       e^ M, N_s
       ^p ]    s
       +
       C τ^β p,

which together with (<ref>), (<ref>)  forces us to recognize from (<ref>) that

    [ | J_1 | ] 
    ≤
    C ∫_0^t 
         [ 
       e^ M, N_s
       ^p ]    s
       +
       C   ( 1N )^ p β 
       +
       C τ^β p.
































































Plugging this and (<ref>), (<ref>)
into (<ref>), taking expectations and

applying the Gronwall inequality give

    e^M,N_t_L^p ( Ω; H) ≤
    C (
         N ^ - β + τ ^β).

Therefore, the desired error bound follows from (<ref>) by taking (<ref>) 
and (<ref>) into account.

□



















 




















































§ NUMERICAL EXPERIMENTS



Some numerical experiments are performed in this section to test the previous theoretical findings.  
Let us consider the 
stochastic Allen-Cahn equation with additive space-time white noise, described by

    {[ ∂ u/∂ t = ∂^2 u/∂ x^2 + u - u^3 +  Ẇ (t),     t ∈ (0, 1],    x ∈ (0,1),;                                    u(0, x) = sin( π x),      x ∈ (0,1),;                                   u(t, 0) = u(t,1) = 0,     t ∈ (0, 1]. ].

Here { W (t) }_t ∈[0, T] is a cylindrical I-Wiener process represented by 
(<ref>).  In what follows, we will use the new fully discrete scheme 
(<ref>) to approximate the continuous problem (<ref>).
Error bounds are always measured in terms of mean-square approximation errors at the endpoint T = 1, 
caused by spatial and temporal discretizations and the expectations are approximated by 
computing averages over 1000 samples.

Before proceeding further with simulations, it is helpful to mention that 
the stochastic convolution in the scheme (<ref>) is easily implementable  
once one realizes that ∫_t_m^t_m+1 E_N(t_m+1-s) P_N  W(s) = 
∑_i=1^N Λ_i e_i, where Λ_i = ∫_t_m^t_m+1 e^   - (t_m+1-s) λ_i β(s), 
1 ≤ i ≤ N are independent, zero-mean normally distributed random variables with explicit variances
[ | Λ_i |^2 ] =  1 - e^ - 2 λ_i τ 2 λ_i. For more details on the implementation
of so-called AEE schemes, one can consult <cit.> and <cit.>.






To visually inspect the convergence rates in space, we identify the “exact” solution by using the full 
discretization with M_exact = N_exact = 2^11 = 2048. 
The spatial approximation errors X(1) - X^N(1) _L^2 ( Ω; H) with
N= 2^i, i = 2, ..., 7 are depicted in Fig.<ref>, against 1N on a log-log scale, where one can observe
that the resulting spatial errors decrease at a slope close to 1/2.
This is consistent with the previous theoretical result (<ref>).












Moreover, we attempt to illustrate the error bound (<ref>) for the fully discrete scheme (<ref>).
As implied by (<ref>), the convergence rate in space is identical to that in time. 
Consequently, we take M = N, p = 2  and β = 1/2 - ϵ
with arbitrarily small ϵ >0 in (<ref>) to arrive at

    X(1) - Y^N,N_t_N_L^2 ( Ω; H) ≤
    C_ϵ 
         N ^ - 1/2 + ϵ .

To see (<ref>), we, similarly as above, do a full  discretization on a very fine mesh with
M_exact = N_exact = 2^11 = 2048 to compute the “exact” solution.
Six different mesh parameters N = 2^i, i = 2,3,...,7 are then used to get six full discretizations. 
The resulting errors are listed in Table <ref> and plotted in Fig.<ref> on a log-log scale. 
From Fig.<ref>,  one can observe the expected convergence rate of order almost 12,
which agrees with that indicated in (<ref>).










   









































 
































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