Convergence of finite element solution of stochastic Burgers equation

: We explore the numerical approximation of the stochastic Burgers equation driven by fractional Brownian motion with Hurst index H ∈ (1 / 4 , 1 / 2) and H ∈ (1 / 2 , 1), respectively. The spatial and temporal regularity properties for the solution are obtained. The given problem is discretized in time with the implicit Euler scheme and in space with the standard finite element method. We obtain the strong convergence of semidiscrete and fully discrete schemes, performing the error estimates on a subset Ω k , h of the sample space Ω with the Gronwall argument being used to overcome the di ffi culties, caused by the subtle interplay of the nonlinear convection term. Numerical examples confirm our theoretical findings.

The stochastic Burgers equation (SBE) has applications in many areas [1][2][3][4][5].For H = 1/2 in (1.1), Bertini et al. [6], Brzeźniak et al. [7], and Kim [8] established the well-posedness results of the SBE.Catuogno and Olivera [9] proved the existence of a strong solution of the SBE.Twardowska and Zabczyk [10] and Goldys and Maslowski [11] obtained the ergodicity results of the SBE.In [12], E et al. established a stationary solution of the SBE.Zou and Wang [13] considered the existence result of a fractional-order SBE.Zhou et al. [14] obtained strong solutions for the SBE.The SBE is a typical evolution equation, and recently a series of numerical methods, like the two-grid method [15][16][17], ADI method [18][19][20], finite difference method [21,22], spectral method [23], finite volume method [24,25], and extrapolation method [26], have been developed to solve it.From a computational point of view, Hairer and Voss [27] devoted their research to the finite difference approximations of the SBE.Hairer and Matetski [28] obtained the optimal convergence rate of the SBE.Blömker and Jentzen [29] studied Galerkin approximations of the SBE.Jentzen et al. [30] proposed and analyzed explicit space-time discrete numerical approximations for the SBE.Uma et al. [31] developed an approximate solution method for solving the SBE.
Fractional Brownian motion (fBm for short) is a type of Gaussian process that exhibits long-range dependence.It is characterized by its self-similarity and its correlation structure which is determined by a parameter called the Hurst index H.It plays a crucial role in various fields.Its applications can be broadly categorized into two main areas: finance and image/signal processing.In finance, fBm is utilized for modeling and simulating price movements in financial markets.It is particularly useful in capturing the long-term dependency and self-similarity observed in asset prices.Furthermore, fBm has notable applications in image and signal processing.Due to its ability to represent fractal-like structures, fBm is employed for texture synthesis and texture modeling, enabling the generation of realistic textures.Moreover, fBm is utilized for noise reduction, interpolation, and denoising in image processing, providing improved image quality and enhanced feature extraction capabilities.These applications have motivated theoretical and numerical investigations of stochastic differential equations (SDE) driven by fBm.For examples of theoretical results of the SDE driven by fBm, see, Wang et al. [32], Jiang et al. [33], Hinz [34], Pei et al. [35], Yang et al. [36], Zou et al. [37], and the references therein.However, there is less literature on studying numerical approximations for the SDE driven by fBm.For example, Cao et al. [38] considered finite element approximations for the SDE driven by fBm.Tudor [39] investigated Wong-Zakai type approximations.Qi and Lin [40] obtained the optimal error bound.Hong et al. [41] investigated the super-convergence result for the stochastic heat equation driven by fBm.The challenge of analyzing the convergence of SDE driven by fBm is that the Burkholder-Davis-Gundy inequality is unavailable.Therefore, it is necessary to take a different strategy to solve this problem.In the existing literature, we are not aware of any numerical investigation of (1.1)- (1.3).This paper aims to fill the gap.
The main difficulty of equations (1.1)-(1.3) is that the noise term causes the solutions u to be very "rough".Thus, new techniques and skills need to be explored.First of all, we establish the following energy estimates where C H,T > 0. Next, we prove the Hölder regularity results, that is, if H ∈ (1/4, 1/2), it holds that and if H ∈ (1/2, 1), we obtain For the detailed definitions of operator A and operator A β/2 , we refer the readers to Section 2. A key tool used here is a very careful treatment of multiple integrals.
To prove the convergence, the main problem here is caused by the subtle interplay of the nonlinear convection term and the stochastic forcing, which prevents a Gronwall argument in the context of expectations.We investigate the error estimates on a subset of Ω and use Markov's inequality to overcome the difficulties.Based on the regularity properties, and for a subset and if H ∈ (1/2, 1), we have where , ε > 0 is arbitrarily small, and k is the time step.For a subset Ω h ⊂ Ω, with P[Ω h ] → 1 as h → 0, where h > 0 is the space step, if H ∈ (1/4, 1/2), we have and for H ∈ (1/2, 1), we get where Ω k,h = Ω k ∩ Ω h , and both ϵ > 0 and ε > 0 are arbitrarily small.It should be emphasized that the convergence analysis presented here requires various delicate error estimates, which is different from the case of the Wiener process.
Section 2 presents some preliminaries.The spatial and temporal regularity results of (1.1)-(1.3)are established in Section 3. Section 4 shows the convergence results for the time discretization scheme.Section 5 is devoted to the space-time discretization.We present the numerical experiments in Section 6.

Notations and preliminaries
We assume that L q (D) and W k,p (D) are Lebesgue and Sobolev spaces.Let L q (D) := [L q (D)] 2 , W k,p (D) := [W k,p (D)] 2 .We define the space We define the operator A via Au = −ν∆u, in the domain where {(γ j , e j )} ∞ j=1 are the pairs of eigenvalues and eigenfunctions of A. Let V s = D(A s 2 ) be the Hilbert space endowed with the norm We define the space L(V) by It is also well-known that (see [42]) and is called an fBm, where H ∈ (0, 1) is the Hurst index.In particular, if H = 1 2 , then the process is in fact Brownian motion or a Wiener process.
Define K H (t, s) by where 1 2 is a constant.By (2.3) and (2.4), we have Let H denote the reproducing kernel Hilbert space of fBm.For every 0 < H < 1, s < τ, we define the linear mapping K * τ : H → L 2 ([0, T ]) [43,44] by where Y and X are real separable Hilbert spaces.
For the definition of the Wiener integral t 0 σ(s)dB H (s), we refer to references [43,45].Then, the stochastic integral satisfies and the inequality holds where {B(t), t ∈ [0, T ]} is a Wiener process, the function σ : [0, T ] → L 0 2 , and the constant C(p) > 0. We introduce the following assumptions: and where C T is a constant. (2.12) The following properties are valid: Using integration by parts, we have From the above estimate, using the Gagliardo-Nirenberg-Sobolev inequality and applying Ladyzhenskaya's inequality ∥u∥ (2.16)More about the above properties of trilinear operator can be found in [46].

The implicit Euler scheme
We let t n = nk, k = T/N, for n = 0, 1, • • • , N, N ∈ N + .Furthermore, we let u 0 := u 0 .We define an implicit Euler scheme by seeking a random variable u n in V such that P-a.s.
where k > 0, and application of Markov's inequality leads to We define the indicator function I Ω k by Using the discrete Gronwall inequality, we have and For any ε > 0, C −1 T ln(k −ε ) > 0, and we denote by and then if 1/4 < H < 1/2, we obtain and for 1/2 < H < 1, it holds that The proofs are finished.

Space-time discretization
We let {T h } 0<h<1 be a regular family of triangulations of D with the maximal mesh size of h.We define the finite element spaces V h by P h is defined as the standard L 2 -projection operator, i.e., we have (see [48]) (5.1) The fully discrete scheme of (1.1)- for any v h ∈ V h , where ∆B H n denotes the fractional Brownian increments.Theorem 5.1.We assume that (S 1 )-(S 2 ) hold.Let u n h and u n be the solutions to (5.2) and (4.1), respectively.For H ∈ (1/4, 1/2), it holds that and if H ∈ (1/2, 1), one can arrive at where Ω h ⊂ Ω such that P[Ω h ] → 1 as h → 0, ϵ > 0 is arbitrarily small.Proof.Setting E n = u n − u n h , from (4.1) and (5.2), we get
Remark.The absence of a nonlinear convective term [u • ∇]u leads to Ω k,h ≡ Ω, that is ε = ϵ = 0 in Theorem 5.2, since the Gronwall inequality may now be utilized directly.We use Newton's iterative algorithm when calculating the nonlinear term in numerical calculations.

Numerical experiments
We present the approximation of fBm B H (t) = t 0 K H (t, s)dB(s), where B(t) is a Brownian motion (see [1]).Let t n = nk for n = 0, 1, . . ., N and k = T/N, then we approximate B H (t) by It should be pointed out that the integral t i+1 t i K H (t n , s)ds is approximated by K H (t n , t i+1 +t i 2 ).The first example provided is to show the convergence rates, and the second example is used to show the numerical simulations about the impacts of fBm on the Burgers equation.u(x, y, 0) = 1, v(x, y, 0) = −1, (x, y) ∈ Ω, u(x, y, t) = 0, v(x, y, t) = 0, (x, y) ∈ ∂Ω, where Ω = {(x, y)|0 ≤ x, y ≤ 2}.We take the temporal interval [0, 1] and the viscosity coefficient ν = 0.2.The errors E[∥e n ∥] := (E[∥u n h −u(t n )∥ 2 ]) 1/2 in the sense of the L 2 -norm are computed by Monte Carlo method over 200 samples, where the "true" solution u(t n ) is approximated by a solution computed by small time step k = 1 160 and space step h = 1 200 .1, if the Hurst index H satisfies 1/4 < H < 1/2, the convergence order is close to the theoretical convergence order O(k min{2H−1/2,H}−ε ), and for Hurst index 1/2 < H < 1, the orders are near to O(k min{3/4,H}−ε ).Table 2 shows that the optimal order of spatial error estimation is consistent with the theoretical result of O(h 1−ϵ ).

Figure 1 .
Figure 1.(A1) and (B1) show the numerical solutions u and v for the corresponding deterministic equation (Ψ(t) = 0).Plots of (A2) and (B2) exhibit the mean values of u and v to the stochastic Burger equation with Hurst index H = 0.3.

Figure 1 (
A1 and B1) shows the numerical results of the corresponding deterministic equation (Ψ(t) = 0).Figure1 (A2 and B2) and Figure2(A3 and B4) exhibit the mean values of u and v to the stochastic Burger equation with Hurst indexes H = 0.3, 0.6, 0.8, respectively.

Table 1 .
Numerical results in the temporal direction with h = 1 200 .