发表机构
Sichuan University; China West Normal University（四川大学; 西华师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带加性迹类噪声的一维随机Burgers方程，对P2有限元空间半离散和全离散格式推导了最优强误差估计及弱收敛率，并给出数值验证。
AI 中文摘要
本文研究带加性迹类噪声的一维粘性随机Burgers方程的有限元逼近。对于$P_2$有限元空间半离散格式，我们在$L^p([0,T] \times \Omega;H_{D}^{\alpha,q})$（其中$p,q\in[2,\infty)$，$\alpha\in[-1,0]$）中推导了关于正则性最优的强误差估计，并在$L^p(\Omega;C([0,T];L^\infty(\mathcal{O})))$中得到了几乎正则性最优的估计。此外，我们推导了终端$L^q$-范数和时空$L^p(0,T;L^q(\mathcal{O}))$-范数的矩的弱误差估计，其弱收敛阶（几乎）是相应强收敛阶的两倍。对于全离散格式，即在空间上采用$P_2$有限元方法、在时间上采用漂移隐式Euler--Maruyama格式，在条件$\tau \leqslant h^2$下，我们在$L^p(\Omega;C([0,T];L^\infty(\mathcal{O})))$的离散类似空间中建立了阶为$\tau^{1/2-\varepsilon}$的强时间收敛率。数值实验验证了理论收敛阶。
英文摘要
This paper investigates finite element approximations of the one-dimensional viscous stochastic Burgers equation with additive trace-class noise. For the $P_2$ finite element spatial semi-discretization, we derive strong error estimates that are optimal with respect to regularity in \(L^p([0,T] \times Ω;H_{D}^{α,q})\) for \(p,q\in[2,\infty)\) and \(α\in[-1,0]\), as well as an almost regularity-optimal estimate in \(L^p(Ω;C([0,T];L^\infty(\mathcal{O})))\). Furthermore, we derive weak error estimates for moments of both terminal $L^q$-norms and space-time $L^p(0,T;L^q(\mathcal{O}))$-norms, with weak convergence rates (nearly) twice the corresponding strong ones. For the fully discrete scheme, which combines the \(P_2\) finite element method in space with a drift-implicit Euler--Maruyama scheme in time, we establish a strong temporal convergence rate of order \(τ^{1/2-\varepsilon}\) in a discrete analogue of \(L^p(Ω;C([0,T];L^\infty(\mathcal{O})))\), under the condition \(τ\leqslant h^2\). Numerical experiments are presented to illustrate the theoretical convergence rates.
