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Binjie Li

Publications and source records attributed to Binjie Li.

At least 19 recordsLinked to original sources
arXiv2026

Convergence of finite element approximations for the one-dimensional stochastic Burgers equation with additive trace-class noise

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.

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arXiv2025

Numerical Analysis of 2D Stochastic Navier--Stokes Equations with Transport Noise: Regularity and Spatial Semidiscretization

This paper establishes strong convergence rates for the spatial finite element discretization of a two-dimensional stochastic Navier--Stokes system with transport noise and no-slip boundary conditions on a convex polygonal domain. The main challenge arises from the lack of spatial \(D(A)\)-regularity of the solution (where \(A\) is the Stokes operator), which prevents the application of standard error analysis techniques. Under a small-noise assumption, we prove that the weak solution satisfies \[ u \in L^2\bigl(Ω; C([0,T]; \dot{H}_σ^{\varrho}) \cap L^2(0,T; \dot{H}_σ^{1+\varrho})\bigr) \] for some \(\varrho \in (0,\tfrac{1}{2})\). To address the low regularity in the numerical analysis, we introduce a novel smoothing operator \(J_{h,α} = A_h^α\mathcal{P}_h A^{-α}\) with \(α\in (0,1)\), where \(A_h\) is the discrete Stokes operator and \(\mathcal{P}_h\) the discrete Helmholtz projection. This tool enables a complete error analysis for a MINI-element spatial semidiscretization, yielding the mean-square convergence estimate \[ \|u - u_h\|_{L^2(Ω; C([0,T]; L^2(\mathcal O;\mathbb{R}^2)))} + \|\nabla(u - u_h)\|_{L^2(Ω\times (0,T); L^2(\mathcal{O};\mathbb{R}^{2\times2}))} \leqslant c\, h^{\varrho} \log\big(1 + \frac{1}{h}\big). \] The framework can be extended to broader stochastic fluid models with rough noise and Dirichlet boundary conditions.

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arXiv2025

Discrete stochastic maximal $ L^p $-regularity and convergence of a spatial semidiscretization for a linear stochastic heat equation

This study investigates the boundedness of the \( H^\infty \)-calculus for the discrete negative Laplace operator, subject to homogeneous Dirichlet boundary conditions. The discrete negative Laplace operator is implemented using the finite element method, and we establish that its \(H^\infty\)-calculus is uniformly bounded with respect to the spatial mesh size. Using this finding, we derive a discrete stochastic maximal \(L^p\)-regularity estimate for a spatial semidiscretization of a linear stochastic heat equation. Furthermore, we provide a nearly optimal pathwise uniform convergence estimate for this spatial semidiscretization within the framework of general spatial \(L^q\)-norms.

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arXiv2025

Convergence of a spatial semidiscretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise

This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial data, the regularity of the mild solution is investigated, and an error estimate is derived within the spatial (L^2)-norm setting. In the case of smooth initial data, two error estimates are established within the framework of general spatial (L^q)-norms.

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arXiv2025

Pathwise uniform convergence of numerical approximations for a two-dimensional stochastic Navier-Stokes equation with no-slip boundary conditions

This paper investigates the pathwise uniform convergence in probability of fully discrete finite-element approximations for the two-dimensional stochastic Navier-Stokes equations with multiplicative noise, subject to no-slip boundary conditions. We demonstrate that the full discretization achieves nearly $ 3/2$-order convergence in space and nearly half-order convergence in time.

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arXiv2024

Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise

This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial approximation. A pathwise uniform convergence rate is derived, encompassing general spatial \( L^q \)-norms, by using discrete versions of deterministic and stochastic maximal \( L^p \)-regularity estimates. Additionally, the theoretical convergence rate is validated through numerical experiments. The primary contribution of this work is the introduction of a technique to establish the pathwise uniform convergence of finite element-based full discretizations for nonlinear stochastic parabolic equations within the framework of general spatial \( L^q \)-norms.

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arXiv2022

Temporal semi-discretizations of a backward semilinear stochastic evolution equation

This paper studies the convergence of three temporal semi-discretizations for a backward semilinear stochastic evolution equation. For general terminal value and general coefficient with Lipschitz continuity, the convergence of the first two temporal semi-discretizations is established, and an explicit convergence rate is derived for the third temporal semi-discretization. The third temporal semi-discretization is applied to a general stochastic linear quadratic control problem, and the convergence of a temporally semi-discrete approximation to the optimal control is established.

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arXiv2022

Convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation

This paper studies the convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semi-discretization uses the standard continuous piecewise linear element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semi-discretization. Thirdly, an application of the theoretical result to a stochastic linear quadratic control problem is presented.

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arXiv2022

Error estimation of a discontinuous Galerkin method for time fractional subdiffusion problems with nonsmooth data

This paper is devoted to the numerical analysis of a piecewise constant discontinuous Galerkin method for time fractional subdiffusion problems. The regularity of weak solution is firstly established by using variational approach and Mittag-Leffler function. Then several optimal error estimates are derived with low regularity data. Finally, numerical experiments are conducted to verify the theoretical results.

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arXiv2022

L1 scheme for solving an inverse problem subject to a fractional diffusion equation

This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle $< π/2 $, that is the resolvent set of this operator contains $ \{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< θ\}$ for some $ π/2 < θ< π$. The relationship between the time fractional order $α\in (0, 1)$ and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as $ α$ approaches $ 1 $. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results.

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arXiv2020

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

This paper is devoted to the numerical analysis of a control constrained distributed optimal control problem subject to a time fractional diffusion equation with non-smooth initial data. The solutions of state and co-state are decomposed into singular and regular parts, and some growth estimates are obtained for the singular parts. Following the variational discretization concept, a full discretization is applied to the corresponding state and co-state equations by using linear conforming finite element method in space and piecewise constant discontinuous Galerkin method in time. By the growth estimates, error estimates are derived with non-smooth initial data. In particular, graded temporal grids are used to obtain the first-order temporal accuracy. Finally, numerical experiments are performed to verify the theoretical results.

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arXiv2020

Temporally semidiscrete approximation of a Dirichlet boundary control for a fractional/normal evolution equation with a final observation

Optimal Dirichlet boundary control for a fractional/normal evolution with a final observation is considered. The unique existence of the solution and the first-order optimality condition of the optimal control problem are derived. The convergence of a temporally semidiscrete approximation is rigorously established, where the control is not explicitly discretized and the state equation is discretized by a discontinuous Galerkin method in time. Numerical results are provided to verify the theoretical results.

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arXiv2020

Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations

Two numerical methods with graded temporal grids are analyzed for fractional evolution equations. One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order $0<α<1$, and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order $1<α<2$. By a new duality technique, pointwise-in-time error estimates of first-order and $ (3-α) $-order temporal accuracies are respectively derived for DG and PG, under reasonable regularity assumptions on the initial value. Numerical experiments are performed to verify the theoretical results.

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arXiv2019

Analysis of the L1 scheme for fractional wave equations with nonsmooth data

This paper analyzes the well-known L1 scheme for fractional wave equations with nonsmooth data. A new stability estimate is obtained, and the temporal accuracy $ \mathcal O(τ^{3-α}) $ is derived for the nonsmooth initial data. In addition, a modified L1 scheme is proposed, and stability and temporal accuracy $ \mathcal O(τ^2) $ are derived for this scheme with nonsmooth initial data. The convergence of the two schemes in the inhomogeneous case is also established. Finally, numerical experiments are performed to verify the theoretical results.

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