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Xiaoping Xie

Publications and source records attributed to Xiaoping Xie.

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 \(L^p([0,T] \times \Omega;H_{D}^{\alpha,q})\) for \(p,q\in[2,\infty)\) and \(\alpha\in[-1,0]\), as well as an almost regularity-optimal estimate in \(L^p(\Omega;C([0,T];L^\infty(\mathcal{O})))\). 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 \(\tau^{1/2-\varepsilon}\) in a discrete analogue of \(L^p(\Omega;C([0,T];L^\infty(\mathcal{O})))\), under the condition \(\tau \leqslant h^2\). Numerical experiments are presented to illustrate the theoretical convergence rates.

math.NA↗
arXiv2026

Physics-informed neural networks for solving two-phase flow problems with moving interfaces

In this paper, a meshfree method using physics-informed neural networks (PINNs) is developed for solving two-phase flow problems with moving interfaces, where two immiscible fluids bearing different material properties, are separated by a dynamically evolving interface and interact with each other through interface conditions. Two kinds of distinct scenarios of interface motion are addressed: the prescribed interface motion whose moving velocity is explicitly given, and the solution-driven interface motion whose evolution is determined by the velocity field of two-phase flow. Based upon piecewise deep neural networks and spatiotemporal sampling points/training set in each fluid subdomain, the proposed PINNs framework reformulates the two-phase flow moving interface problem as a least-squares (LS) minimization problem, which involves all residuals of governing equations, interface conditions, boundary conditions and initial conditions. Furthermore, approximation properties of the proposed PINNs approach are analyzed rigorously for the presented two-phase flow model by employing the Reynolds transport theorem in evolving domains, moreover, a comprehensive error estimation is provided to account for additional complexities introduced by the moving interface and the coupling between fluid dynamics and interface evolution. Numerical experiments are carried out to illustrate the effectiveness of the proposed PINNs approach for various configurations of two-phase flow moving interface problems, and to validate the theoretical findings as well. A practical guidance is thus provided for an efficient training set distribution when applying the proposed PINNs approach to two-phase flow moving interface problems in practice.

math.NA↗
arXiv2026

Stability Analysis of Monolithic Globally Divergence-Free ALE-HDG Methods for Fluid-Structure Interaction

In this paper, we propose two monolithic fully discrete finite element methods for fluid-structure interaction (FSI) based on a novel Piola-type Arbitrary Lagrangian-Eulerian (ALE) mapping. For the temporal discretization, we apply the backward Euler method to both the non-conservative and conservative formulations. For the spatial discretization, we adopt arbitrary order hybridizable discontinuous Galerkin (HDG) methods for the incompressible Navier-Stokes and linear elasticity equations, and a continuous Galerkin (CG) method for the fluid mesh movement. We derive stability results for both the temporal semi-discretization and the fully discretization, and show that the velocity approximations of the fully discrete schemes are globally divergence-free. Several numerical experiments are performed to verify the performance of the proposed methods.

math.NA↗
arXiv2025

Analysis of a fast fully discrete finite element method for fractional viscoelastic wave propagation

This paper is devoted to a numerical analysis of a fractional viscoelastic wave propagation model that generalizes the fractional Maxwell model and the fractional Zener model. First, we convert the model problem into a velocity type integro-differential equation and establish existence, uniqueness and regularity of its solution. Then we consider a conforming linear/bilinear/trilinear finite element semi-discrete scheme and a fast scheme of backward Euler full discretization with a sum-of-exponentials (SOE) approximation for the convolution integral, and derive error estimates for the semi-discrete and fully discrete schemes. Finally, we provide several numerical examples to verify the theoretical results.

math.NA↗
arXiv2025

A modified dynamic diffusion finite element method with optimal convergence rate for convection-diffusion-reaction equations

In this paper, we develop a modified nonlinear dynamic diffusion (DD) finite element method for convection-diffusion-reaction equations. This method is free of stabilization parameters and is capable of precluding spurious oscillations. We prove existence and, under an assumption of small mesh size, uniqueness of the discrete solution, and derive the optimal first order convergence rate of the approximation error in the energy norm plus a dissipation term. Numerical examples are provided to verify the theoretical analysis.

math.NA↗
arXiv2024

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.

math.NA↗
arXiv2024

Energy-stable mixed finite element methods for the Rosensweig ferrofluid flow model

In this paper, we consider mixed finite element semi-/full discretizations of the Rosensweig ferrofluid flow model. We first establish some regularity results for the model under several basic assumptions. Then we show that the energy stability of the weak solutions is preserved exactly for both the semi-discrete and fully discrete finite element solutions. Moreover, we prove the existence and uniqueness of the discrete solutions. We also derive optimal error estimates for the discrete schemes. Finally, we provide numerical experiments to verify the theoretical results.

math.NA↗
arXiv2024

Robust globally divergence-free HDG finite element method for steady thermally coupled incompressible MHD flow

This paper develops an hybridizable discontinuous Galerkin (HDG) finite element method of arbitrary order for the steady thermally coupled incompressible Magnetohydrodynamics (MHD) flow. The HDG scheme uses piecewise polynomials of degrees $k(k\geq 1),k,k-1,k-1$, and $k$ respectively for the approximations of the velocity, the magnetic field, the pressure, the magnetic pseudo-pressure, and the temperature in the interior of elements, and uses piecewise polynomials of degree $k$ for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. Existence and uniqueness results for the discrete scheme are given and optimal a priori error estimates are derived. Numerical experiments are provided to verify the obtained theoretical results.

math.NA↗
arXiv2024

Robust globally divergence-free weak Galerkin methods for unsteady incompressible convective Brinkman-Forchheimer equations

This paper develops and analyzes a class of semi-discrete and fully discrete weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations. For the spatial discretization, the methods adopt the piecewise polynomials of degrees $m\ (m\geq1)$ and $m-1$ respectively to approximate the velocity and pressure inside the elements, and piecewise polynomials of degree $m$ to approximate their numerical traces on the interfaces of elements. In the fully discrete method, the backward Euler difference scheme is used to approximate the time derivative. The methods are shown to yield globally divergence-free velocity approximation. Optimal a priori error estimates in the energy norm and $L^2$ norm are established. A convergent linearized iterative algorithm is designed for solving the fully discrete system. Numerical experiments are provided to verify the theoretical results.

math.NA↗
arXiv2024

A fast fully discrete mixed finite element scheme for fractional viscoelastic models of wave propagation

Due to the nonlocal feature of fractional differential operators, the numerical solution to fractional partial differential equations usually requires expensive memory and computation costs. This paper develops a fast scheme for fractional viscoelastic models of wave propagation. We first apply the Laplace transform to convert the time-fractional constitutive equation into an integro-differential form that involves the Mittag-Leffler function as a convolution kernel. Then we construct an efficient sum-of-exponentials (SOE) approximation for the Mittag-Leffler function. We use mixed finite elements for the spatial discretization and the Newmark scheme for the temporal discretization of the second time-derivative of the displacement variable in the kinematical equation and finally obtain the fast algorithm. Compared with the traditional L1 scheme for time fractional derivative, our fast scheme reduces the memory complexity from $\mathcal O(N_sN) $ to $\mathcal O(N_sN_{exp})$ and the computation complexity from $\mathcal O(N_sN^2)$ to $\mathcal O(N_sN_{exp}N)$, where $N$ denotes the total number of temporal grid points, $N_{exp}$ the number of exponentials in SOE, and $N_s$ the complexity of memory and computation related to the spatial discretization. Numerical experiments confirm the theoretical results.

math.NA↗
arXiv2024

A Projection-Based Time-Segmented Reduced Order Model for Fluid-Structure Interactions

In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian--Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (horizontal/vertical) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.

cs.CE↗
arXiv2023

Robust globally divergence-free Weak Galerkin finite element method for incompressible Magnetohydrodynamics flow

This paper develops a weak Galerkin (WG) finite element method of arbitrary order for the steady incompressible Magnetohydrodynamics equations. The WG scheme uses piecewise polynomials of degrees $k(k\geq 1),k,k-1$, and $k-1$ respectively for the approximations of the velocity, the magnetic field, the pressure, and the magnetic pseudo-pressure in the interior of elements, and uses piecewise polynomials of degree $k$ for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. We give existence and uniqueness results for the discrete scheme and derive optimal a priori error estimates. We also present a convergent linearized iterative algorithm. Numerical experiments are provided to verify the obtained theoretical results.

math.NA↗
arXiv2023

A new error analysis for parabolic Dirichlet boundary control problems

In this paper, we consider the finite element approximation to a parabolic Dirichlet boundary control problem and establish new a priori error estimates. In the temporal semi-discretization we apply the DG(0) method for the state and the variational discretization for the control, and obtain the convergence rates $O(k^{\frac{1}{4}})$ and $O(k^{\frac{3}{4}-\varepsilon})$ $(\varepsilon>0)$ for the control for problems posed on polytopes with $y_0\in L^2(Ω)$, $y_d\in L^2(I;L^2(Ω))$ and smooth domains with $y_0\in H^{\frac{1}{2}}(Ω)$, $y_d\in L^2(I;H^1(Ω))\cap H^{\frac{1}{2}}(I;L^2(Ω))$, respectively. In the fully discretization of the optimal control problem posed on polytopal domains, we apply the DG(0)-CG(1) method for the state and the variational discretization approach for the control, and derive the convergence order $O(k^{\frac{1}{4}} +h^{\frac{1}{2}})$, which improves the known results by removing the mesh size condition $k=O(h^2)$ between the space mesh size $h$ and the time step $k$. As a byproduct, we obtain a priori error estimate $O(h+k^{1\over 2})$ for the fully discretization of parabolic equations with inhomogeneous Dirichlet data posed on polytopes, which also improves the known error estimate by removing the above mesh size condition.

math.NA↗
arXiv2023

A Voice Disease Detection Method Based on MFCCs and Shallow CNN

The incidence rate of voice diseases is increasing year by year. The use of software for remote diagnosis is a technical development trend and has important practical value. Among voice diseases, common diseases that cause hoarseness include spasmodic dysphonia, vocal cord paralysis, vocal nodule, and vocal cord polyp. This paper presents a voice disease detection method that can be applied in a wide range of clinical. We cooperated with Xiangya Hospital of Central South University to collect voice samples from sixty-one different patients. The Mel Frequency Cepstrum Coefficient (MFCC) parameters are extracted as input features to describe the voice in the form of data. An innovative model combining MFCC parameters and single convolution layer CNN is proposed for fast calculation and classification. The highest accuracy we achieved was 92%, it is fully ahead of the original research results and internationally advanced. And we use Advanced Voice Function Assessment Databases (AVFAD) to evaluate the generalization ability of the method we proposed, which achieved an accuracy rate of 98%. Experiments on clinical and standard datasets show that for the pathological detection of voice diseases, our method has greatly improved in accuracy and computational efficiency.

cs.SD↗
arXiv2023

A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration

In this paper, we develop a new reduced basis (RB) method, named as Single Eigenvalue Acceleration Method (SEAM), for second-order parabolic equations with homogeneous Dirichlet boundary conditions. The high-fidelity numerical method adopts the backward Euler scheme and conforming finite elements for the temporal and spatial discretization, respectively. Under the assumption that the time step size is sufficiently small and time steps are not very large, we show that the singular value distribution of the high-fidelity solution matrix $U$ is close to that of a rank one matrix. We select the eigenfunction associated with the principal eigenvalue of the matrix $U^\top U$ as the basis of the Proper Orthogonal Decomposition (POD) method to obtain SEAM and a parallel SEAM. Numerical experiments confirm the efficiency of the new method.

math.NA↗
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.

math.NA↗