arXiv subjects

Ruisheng Qi

Publications and source records attributed to Ruisheng Qi.

9 recordsLinked to original sources

arXiv2026

Long-time error analysis of finite element fully discrete schemes for SPDEs with non-globally Lipschitz coefficients

The present paper proposes new fully discrete schemes for long-time approximations of stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients in a bounded domain $D \subset \R^d, d =1,2,3 $. A novel family of linearly implicit time-stepping schemes is introduced, based on a standard Galerkin finite element spatial semi-discretization. A distinguishing feature of the schemes is that the proposed finite element fully discrete approximations preserve uniform-in-time moment bounds in a Banach space $L^{r}(D), r >2$, without requiring any restriction on the time-space discretization stepsize ratio. %established... To show it, some non-standard arguments are developed. First, we derive long-time error estimates in the Banach space $L^r(D)$ for finite element fully discrete approximations of the deterministic linear parabolic equation with non-smooth initial value, which is, to our knowledge, new for the literature on numerical PDEs and of independent interest. These error estimates together with the contractive property of the semi-group in $L^{r}(D), r > 2$, the dissipativity of the nonlinearity and the particular benefit of the taming strategy help us establish the desired uniform-in-time moment bounds. Then both strong and weak error bounds of the proposed schemes are carefully analyzed in a setting of low regularity, with uniform-in-time convergence rates obtained for cases of both space-time white and trace-class noises. The analysis is highly nontrivial, due to the finite element discretization, the low regularity and the presence of the super-linearly growing nonlinearity. %in the long-time scenario... the discretization parameters $h$ and $τ$. Finally, numerical results are presented to verify the previous theoretical findings.

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arXiv2024

Strong convergence rates of Galerkin finite element methods for SWEs with cubic polynomial nonlinearity

In the present work, strong approximation errors are analyzed for both the spatial semi-discretization and the spatio-temporal fully discretization of stochastic wave equations (SWEs) with cubic polynomial nonlinearities and additive noises. The fully discretization is achieved by the standard Galerkin ffnite element method in space and a novel exponential time integrator combined with the averaged vector ffeld approach. The newly proposed scheme is proved to exactly satisfy a trace formula based on an energy functional. Recovering the convergence rates of the scheme, however, meets essential difffculties, due to the lack of the global monotonicity condition. To overcome this issue, we derive the exponential integrability property of the considered numerical approximations, by the energy functional. Armed with these properties, we obtain the strong convergence rates of the approximations in both spatial and temporal direction. Finally, numerical results are presented to verify the previously theoretical findings.

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arXiv2023

Strong convergence rates of an explicit scheme for stochastic Cahn--Hilliard equation with additive noise

In this paper, we propose and analyze an explicit time-stepping scheme for a spatial discretization of stochastic Cahn--Hilliard equation with additive noise. The fully discrete approximation combines a spectral Galerkin method in space with a tamed exponential Euler method in time. In contrast to implicit schemes in the literature, the explicit scheme here is easily implementable and produces significant improvement in the computational efficiency. It is shown that the fully discrete approximation converges strongly to the exact solution, with strong convergence rates identified. Different from the tamed time-stepping schemes for stochastic Allen--Cahn equations, essential difficulties arise in the analysis due to the presence of the unbounded linear operator in front of the nonlinearity. To overcome them, new and non-trivial arguments are developed in the present work. To the best of our knowledge, it is the first result concerning an explicit scheme for the stochastic Cahn--Hilliard equation. Numerical experiments are finally performed to confirm the theoretical results.

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arXiv2022

Strong convergence rates of a fully discrete scheme for the Cahn-Hilliard-Cook equation

The first aim of this paper is to examine existence, uniqueness and regularity for the Cahn-Hilliard-Cook (CHC) equation in space dimension $d\leq 3$. By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem {\color{black}{with additive noise}} into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo-Nirenberg inequality and done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original CHC equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results of the problem help us to identify strong convergence rates of the fully discrete scheme.

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arXiv2020

Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise

Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive noise in space dimension $d \leq 3$. The full discretization is realized by combining the standard finite element method with the backward Euler time-stepping scheme. Distinct from the globally Lipschitz setting, the error analysis becomes rather challenging and demanding, due to the presence of the cubic nonlinearity in the underlying model. By introducing two auxiliary approximation processes, we propose an appropriate decomposition of the considered error terms and introduce a novel approach of error analysis, to successfully recover the convergence rates of the numerical schemes. The approach is original and does not rely on high-order spatial regularity properties of the approximation processes. It is shown that the fully discrete scheme possesses convergence rates of order $ O(h^γ ) $ in space and order $ O( τ^{ \fracγ{2} } )$ in time, subject to the spatial correlation of the noise process, characterized by $ \|A^{\frac{γ-1}2}Q^{\frac12}\|_{\mathcal{L}_2}<\infty, \, γ\in[\frac d3,2] $, $ d\in\{1,2,3\}$. In particular, a classical convergence rate of order $O(h^2 +τ)$ is reachable, even in multiple space dimensions, when the aforementioned condition is fulfilled with $ γ= 2 $. Numerical examples confirm the previous findings.

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arXiv2020

Error estimates of semi-discrete and fully discrete finite element methods for the Cahn-Hilliard-Cook equation

In two recent publications [Kov{á}cs, Larsson, and Mesforush, SIAM J. Numer. Anal. 49(6), 2407-2429, 2011] and [Furihata, et al., SIAM J. Numer. Anal. 56(2), 708-731, 2018], strong convergence of the semi-discrete and fully discrete finite element methods is, respectively, proved for the Cahn-Hilliard-Cook (CHC) equation, but without convergence rates revealed. The present work aims to fill the left gap, by recovering strong convergence rates of (fully discrete) finite element methods for the CHC equation. More accurately, strong convergence rates of a full discretization are obtained, based on Galerkin finite element methods for the spatial discretization and the backward Euler method for the temporal discretization. It turns out that the convergence rates heavily depend on the spatial regularity of the noise process. Different from the stochastic Allen-Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity in the underlying model makes the error analysis much more challenging and demanding. To address such difficulties, several new techniques and error estimates are developed. Numerical examples are finally provided to confirm the previous findings.

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arXiv2016

Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations

This article offers sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater than one-half. In addition, mean-square numerical approximation of such problem are investigated, performed by the spectral Galerkin method in space and the linear implicit Euler method in time. The obtained sharp regularity properties of the problems enable us to identify optimal mean-square convergence rates of the full discrete scheme. These theoretical findings are accompanied by several numerical examples.

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arXiv2016

An accelerated exponential time integrator for semi-linear stochastic strongly damped wave equation with additive noise

This paper is concerned with the strong approximation of a semi-linear stochastic wave equation with strong damping, driven by additive noise. Based on a spatial discretization performed by a spectral Galerkin method, we introduce a kind of accelerated exponential time integrator involving linear functionals of the noise. Under appropriate assumptions, we provide error bounds for the proposed full-discrete scheme. It is shown that the scheme achieves higher strong order in time direction than the order of temporal regularity of the underlying problem, which allows for higher convergence rate than usual time-stepping schemes. For the space-time white noise case in two or three spatial dimensions, the scheme still exhibits a good convergence performance. Another striking finding is that, even for the velocity with low regularity the scheme always promises first order strong convergence in time. Numerical examples are finally reported to confirm our theoretical findings.

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arXiv2015

Error estimates of finite element method for semi-linear stochastic strongly damped wave equation

In this paper, we consider a semi-linear stochastic strongly damped wave equation driven by additive Gaussian noise. Following a semigroup framework, we establish existence, uniqueness and space-time regularity of a mild solution to such equation. Unlike the usual stochastic wave equation without damping, the underlying problem with space-time white noise (Q = I) allows for a mild solution with a positive order of regularity in multiple spatial dimensions. Further, we analyze a spatio-temporal discretization of the problem, performed by a standard finite element method in space and a well-known linear implicit Euler scheme in time. The analysis of the approximation error forces us to significantly enrich existing error estimates of semidiscrete and fully discrete finite element methods for the corresponding linear deterministic equation. The main results show optimal convergence rates in the sense that the orders of convergence in space and in time coincide with the orders of the spatial and temporal regularity of the mild solution, respectively. Numerical examples are finally included to confirm our theoretical findings.

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