Convergence of finite element approximations for the one-dimensional stochastic Burgers equation with additive trace-class noise
Abstract
This paper investigates finite element approximations of the one-dimensional viscous stochastic Burgers equation with additive trace-class noise. For the P2 finite element spatial semi-discretization, we derive strong error estimates that are optimal with respect to regularity in Lp([0,T] × Ω;HDα,q) for p,q∈[2,∞) and α∈[−1,0], as well as an almost regularity-optimal estimate in Lp(Ω;C([0,T];L∞(𝒪))). Furthermore, we derive weak error estimates for moments of both terminal Lq-norms and space-time Lp(0,T;Lq(𝒪))-norms, with weak convergence rates (nearly) twice the corresponding strong ones. For the fully discrete scheme, which combines the P2 finite element method in space with a drift-implicit Euler--Maruyama scheme in time, we establish a strong temporal convergence rate of order τ1/2−ε in a discrete analogue of Lp(Ω;C([0,T];L∞(𝒪))), under the condition τleqslant h2. Numerical experiments are presented to illustrate the theoretical convergence rates.
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BibTeX
@misc{li2026convergence,
title = {{Convergence of finite element approximations for the one-dimensional stochastic Burgers equation with additive trace-class noise}},
author = {Binjie Li and Xiaoping Xie and Qin Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11584},
eprint = {2609.11584},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11584},
}
RIS
TY - UNPB
TI - Convergence of finite element approximations for the one-dimensional stochastic Burgers equation with additive trace-class noise
AU - Binjie Li
AU - Xiaoping Xie
AU - Qin Zhou
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11584
UR - https://arxiv.org/abs/2609.11584
AB - 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] × Ω;H_D^α,q) for p,q∈[2,∞) and α∈[−1,0], as well as an almost regularity-optimal estimate in L^p(Ω;C([0,T];L^∞(𝒪))). Furthermore, we derive weak error estimates for moments of both terminal L^q-norms and space-time L^p(0,T;L^q(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−ε in a discrete analogue of L^p(Ω;C([0,T];L^∞(𝒪))), under the condition τleqslant h^2. Numerical experiments are presented to illustrate the theoretical convergence rates.
ER -