arXiv subjects

Yongkui Zou

Publications and source records attributed to Yongkui Zou.

6 recordsLinked to original sources

arXiv2026

Locking analysis and high-order locking-free finite element method for three-dimensional electroporoelasticity equations

Electroporoelasticity equations couple Maxwell's equations with Biot's poroelasticity model and admit severe Poisson locking in standard conforming finite element discretizations when the Lam\'e constant is large. In this paper, we provide a rigorous analysis of the Poisson locking phenomenon for three-dimensional quasi-static electroporoelasticity equations and show that the spatial convergence order of conforming finite element approximations is reduced in the nearly incompressible regime. To eliminate this locking effect, we introduce a five-field formulation and a fully discrete high-order finite element method. We prove uniform stability with respect to large Lam\'e constant. Based on this, we establish a locking-free scheme by deriving its uniform error estimates with respect to the Lam\'e coefficient. The analysis covers the fully coupled electromagnetic-poroelastic system and applies to high-order elements in three dimensions. Extensive numerical experiments are presented to verify the theoretical convergence rates and demonstrate robustness with respect to the Lam\'e parameter.

math.NA↗

arXiv2025

Existence and smoothness of density function of solution to Mckean--Vlasov Equation with general coefficients

In this paper, we study the existence and smoothness of a density function to the solution of a Mckean-Vlasov equation with the aid of Malliavin calculus. We first show the existence of the density function under assumptions that the coefficients of equation are only Lipschitz continuity and satisfy a uniform elliptic condition. Furthermore, we derive a precise regularity order and bounded a priori estimate for the density function under optimal smoothness assumptions for the coefficients. Finally, we present several numerical experiments to illustrate the approximation of the density function independently determined by solving a Fokker-Planck equation.

math.AP↗

arXiv2025

A novel numerical method for mean field stochastic differential equation

In this paper, we propose a novel method to approximate the mean field stochastic differential equation by means of approximating the density function via Fokker-Planck equation. We construct a well-posed truncated Fokker-Planck equation whose solution is an approximation to the density function of solution to the mean field stochastic differential equation. We also apply finite difference method to approximate the truncated Fokker-Planck equation and derive error estimates. We use the numerical density function to replace the true measure in mean field stochastic differential equation and set up a stochastic differential equation to approximate the mean field one. Meanwhile, we derive the corresponding error estimates. Finally, we present several numerical experiments to illustrate the theoretical analysis.

math.NA↗

arXiv2025

Weak Convergence Analysis for the Finite Element Approximation to Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise

In this paper, we aim to study the optimal weak convergence order for the finite element approximation to a stochastic Allen-Cahn equation driven by multiplicative white noise. We first construct an auxiliary equation based on the splitting-up technique and derive prior estimates for the corresponding Kolmogorov equation and obtain the strong convergence order of 1 in time between the auxiliary and exact solutions. Then, we prove the optimal weak convergence order of the finite element approximation to the stochastic Allen-Cahn equation by deriving the weak convergence order between the finite element approximation and the auxiliary solution via the theory of Kolmogorov equation and Malliavin calculus. Finally, we present a numerical experiment to illustrate the theoretical analysis.

math.NA↗

arXiv2025

Splitting finite element approximations for quasi-static electroporoelasticity equations

The electroporoelasticity model, which couples Maxwell's equations with Biot's equations, plays a critical role in applications such as water conservancy exploration, earthquake early warning, and various other fields. This work focuses on investigating its well-posedness and analyzing error estimates for a splitting backward Euler finite element method. We first define a weak solution consistent with the finite element framework. Then, we prove the uniqueness and existence of such a solution using the Galerkin method and derive a priori estimates for high-order regularity. Using a splitting technique, we define an approximate splitting solution and analyze its convergence order. Next, we apply Nedelec's curl-conforming finite elements, Lagrange elements, and the backward Euler method to construct a fully discretized scheme. We demonstrate the stability of the splitting numerical solution and provide error estimates for its convergence order in both temporal and spatial variables. Finally, we present numerical experiments to validate the theoretical results, showing that our method significantly reduces computational complexity compared to the classical finite element method.

math.NA↗

arXiv2022

Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is a unnormalized probability density function of the filter solution. Then we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.

math.NA↗