arXiv subjects

Shuaishuai Lu

Publications and source records attributed to Shuaishuai Lu.

7 recordsLinked to original sources

arXiv2026

Polynomial preserving recoveries of edge element method on Cartesian grids for the time-harmonic Maxwell equations with large wave number

This paper considers the lowest-order first type Nédélec edge element method (EEM) on Cartesian grids for the three-dimensional time-harmonic Maxwell equations with a large wave number. New polynomial preserving recovery (PPR) operators are proposed for the curl of the edge element solution and for the solution itself, respectively. Under the condition that $κ^3 h^2 C_{\mathrm{sol}}$ is sufficiently small, second-order superconvergence estimates are proved for both the recovered curl and the recovered solution, where $κ$ is the wave number, $h$ is the mesh size, and $C_{\mathrm{sol}}$ is a stability constant associated with the Maxwell solution operator. In particular, the analysis shows that the proposed PPR procedures cannot mitigate the well-known pollution effect inherent to the EEM. To reduce the pollution error, we further propose a new continuous interior penalty edge element method (CIP-EEM) that incorporates an additional normal-jump penalty term. It is shown that by appropriately choosing the penalty parameters, the new CIP-EEM can improve the phase error by two orders in $κh$. Numerical experiments are presented to confirm the theoretical superconvergence results and to demonstrate that the CIP-EEM can effectively reduce the pollution error in the high-frequency regime.

math.NA↗

arXiv2025

Preasymptotic error estimates of higher-order EEM for the time-harmonic Maxwell equations with large wave number

The time-harmonic Maxwell equations with impedance boundary condition and large wave number are discretized using the second-type Nédélec's edge element method (EEM). Preasymptotic error bounds are derived, showing that, under the mesh condition $κ^{2p+1}h^{2p}$ being sufficiently small, the error of the EEM of order $p$ in the energy norm is bounded by $\mathcal{O}\big(κ^{p}h^p + κ^{2p+1}h^{2p}\big)$, while the error in the $κ$-scaled $\boldsymbol{L}^2$ norm is bounded by $\mathcal{O}\big((κh)^{p+1} + κ^{2p+1} h^{2p}\big)$. Here, $κ$ is the wave number and $h$ is the mesh size. Numerical tests are provided to illustrate our theoretical results.

math.NA↗

arXiv2025

The weak averaging principle of stochastic functional partial differential equations with H$\ddot{\text{o}}$lder continuous coefficients and infinite delay

In this paper, we establish the weak averaging principle for stochastic functional partial differential equations (in short, SFPDEs) with H$\ddot{\text{o}}$lder continuous coefficients and infinite delay by a new generalized coupling approach. Firstly, we rigorously establish the existence and uniqueness of weak solutions for a specific class of finite-dimensional systems by the generalized coupling approach. Then we extend these results to their infinite-dimensional counterparts using the variational approach and Galerkin projection technique. Subsequently, we establish the averaging principle for SFPDEs with infinite delay in the weak sense, i.e., we prove that the solution of the original system converges in law to that of the averaged system on a finite interval $[0,T]$ as the small parameter $\varepsilon\to 0$. To illustrate our findings, we present two applications: stochastic generalized porous media equations and stochastic reaction-diffusion equations.

math.PR↗

arXiv2025

McKean-Vlasov SPDEs with coefficients exhibiting locally weak monotonicity: existence, uniqueness, ergodicity, exponential mixing and limit theorems

This paper investigates the existence and uniqueness of solutions, as well as the ergodicity and exponential mixing to invariant measures, and limit theorems for a class of McKean-Vlasov SPDEs with locally weak monotonicity. In particular, for a class of weak monotonicity conditions, including H$\ddot{\text{o}}$lder continuity, we rigorously establish the existence and uniqueness of weak solutions to McKean-Vlasov SPDEs by employing the Galerkin projection technique and the generalized coupling approach. Additionally, we explore the properties of the solutions, including time homogeneity, the Markov and the Feller property. Building upon these properties, we examine the exponential ergodicity and mixing of invariant measures under Lyapunov conditions. Finally, within the framework of coefficients meeting the criteria of locally weak monotonicity and Lyapunov conditions, alongside the uniform mixing property of invariant measures, we establish the strong law of large numbers and the central limit theorem for the solution and obtain estimates of corresponding convergence rates.

math.PR↗

arXiv2025

Stochastic tamed 3D Navier-Stokes equations with locally weak monotonicity coefficients: existence, uniqueness and averaging principle

This paper investigates the stochastic tamed 3D Navier-Stokes equations with locally weak monotonicity coefficients in the whole space as well as in the three-dimensional torus, which play a crucial role in turbulent flows analysis. A significant issue is addressed in this work, specifically, the reduced regularity of the coefficients and the inapplicability of Gronwall's lemma complicates the establishment of pathwise uniqueness for weak solutions. Initially, the existence of a martingale solution for the system is established via Galerkin approximation; thereafter, the pathwise uniqueness of this martingale solution is confirmed by constructing a specialized control function. Ultimately, the Yamada-Watanabe theorem is employed to establish the existence and uniqueness of the strong solution to the system. Furthermore, an averaging principle, referred to as the first Bogolyubov theorem, is established for stochastic tamed 3D Navier-Stokes equations with highly oscillating components, where the coefficients satisfy the assumptions of linear growth and locally weak monotonicity. This result is achieved using classical Khasminskii time discretization, which illustrates the convergence of the solution from the original Cauchy problem to the averaged equation over a finite interval [0, T].

math.PR↗

arXiv2024

Preasymptotic error estimates of EEM and CIP-EEM for the time-harmonic Maxwell equations with large wave number

Preasymptotic error estimates are derived for the linear edge element method (EEM) and the linear $\boldsymbol{H}(\boldsymbol{\mathrm{curl}})$-conforming interior penalty edge element method (CIP-EEM) for the time-harmonic Maxwell equations with large wave number. It is shown that under the mesh condition that $κ^3 h^2$ is sufficiently small, the errors of the solutions to both methods are bounded by $\mathcal{O} (κh + κ^3 h^2 )$ in the energy norm and $\mathcal{O} (κh^2 + κ^2 h^2 )$ in the $\boldsymbol{L}^2$ norm, where $κ$ is the wave number and $h$ is the mesh size. Numerical tests are provided to verify our theoretical results and to illustrate the potential of CIP-EEM in significantly reducing the pollution effect.

math.NA↗

arXiv2024

Stochastic functional partial differential equations with monotone coefficients: Poisson stability measures, exponential mixing and limit theorems

This paper examines Poisson stable (including stationary, periodic, almost periodic, Levitan almost periodic, Bohr almost automorphic, pseudo-periodic, Birkhoff recurrent, pseudo-recurrent, etc.) measures and limit theorems for stochastic functional partial differential equations(SFPDEs) with monotone coefficients. We first show the existence and uniqueness of entrance measure $μ_{t}$ for SFPDEs by dissipative method (or remoting start). Then, with the help of Shcherbakov's comparability method in character of recurrence, we prove that the entrance measure inherits the same recurrence of coefficients. Thirdly, we show the tightness of the set of measures $μ_{t}$. As a result, any sequence of the average of $\{μ_{t}\}_{t\in\mathbb{R} }$ have the limit point $μ^{*}$. Further, we study the uniform exponential mixing of the measure $μ^{*}$ in the sense of Wasserstein metric. Fourthly, under uniform exponential mixing and Markov property, we establish the strong law of large numbers, the central limit theorem and estimate the corresponding rates of convergence for solution maps of SFPDEs. Finally, we give applications of stochastic generalized porous media equations with delay to illustrate of our results.

math.PR↗