arXiv subjects

Yufeng Nie

Publications and source records attributed to Yufeng Nie.

At least 19 recordsLinked to original sources

arXiv2026

A nonlocal nonlinear Schrödinger model: well-posedness, local limit, and structure-preserving asymptotically compatible Fourier approximations

In this paper, we introduce a nonlocal nonlinear Schrödinger (NLS) model on periodic domains, and establish its well-posedness, conservation laws, local limit, dispersion properties, and develop a structure-preserving asymptotically compatible Fourier collocation method. We prove that for a fixed nonlocal horizon $δ$, the model is globally well posed and conserves mass and nonlocal energy, and the nonlocal NLS solution converges to the local NLS solution with order $O(δ^2)$ under suitable regularity assumptions. We also derive the dispersion relation and group velocity for plane waves, establish their $O(δ^2)$ local limits, and characterize their behavior at high frequencies. The numerical method combines Crank--Nicolson time stepping with Fourier collocation and preserves mass and energy. The existence, uniqueness, and convergence of the numerical solutions are proved. In particular, the error of the nonlocal NLS solution is $O(τ^2+N^{s-r})$ in the $H^s$ norm, uniformly with respect to the horizon. Moreover, its total $H^s$-error relative to the local NLS solution is $O(δ^2+τ^2+N^{s-r})$ without any coupling condition among $δ$, $τ$, and $N$, which proves asymptotic compatibility of the proposed method. Numerical experiments in one, two, and three dimensions are presented to verify the theoretical accuracy and discrete conservation, confirm convergence under independent variation of horizon and discretization parameters, and show how the horizon and kernel affect dispersive wave propagation.

math.NA↗

arXiv2025

Higher-order multi-scale computational method and its convergence analysis for hygro-thermo-mechanical coupling problems of quasi-periodic composite structures

This paper proposes a novel higher-order multi-scale (HOMS) computational method, which is highly targeted for efficient, high-accuracy and low-computational-cost simulation of hygro-thermo-mechanical (H-T-M) coupling problems in quasi-periodic composite structures. The first innovation of this work is that the establishment of the high-accuracy multi-scale model incorporating the higher-order correction terms for H-T-M coupling problems of quasi-periodic composite structures. The second innovation of this work is that the error analyses in the point-wise and integral senses are rigorously derived for multi-scale asymptotic solutions. Especially from the point-wise error analysis, the primary impetus for current study to develop the HOMS approach for quasi-periodic composite structures is illustrated. Furthermore, an high-accuracy multi-scale numerical algorithm is developed based on finite element method, while corresponding convergent analysis is also obtained. Finally, extensive numerical experiments are conducted to validate the computational performance of the proposed HOMS computational approach, demonstrating not only exceptional numerical accuracy, but also reduced computational cost.

math.NA↗

arXiv2025

A Spectral Localization Method for Time-Fractional Integro-Differential Equations with Nonsmooth Data

In this work, we develop a localized numerical scheme with low regularity requirements for solving time-fractional integro-differential equations. First, a fully discrete numerical scheme is constructed. Specifically, for temporal discretization, we employ the contour integral method (CIM) with parameterized hyperbolic contours to approximate the nonlocal operators. For spatial discretization, the standard piecewise linear Galerkin finite element method (FEM) is used. We then provide a rigorous error analysis, demonstrating that the proposed scheme achieves high accuracy even for problems with nonsmooth/vanishing initial values or low-regularity solutions, featuring spectral accuracy in time and second-order convergence in space. Finally, a series of numerical experiments in both 1-D and 2-D validate the theoretical findings and confirm that the algorithm combines the advantages of spectral accuracy, low computational cost, and efficient memory usage.

math.NA↗

arXiv2025

SO-PIFRNN: Self-optimization physics-informed Fourier-features randomized neural network for solving partial differential equations

This study proposes a self-optimization physics-informed Fourier-features randomized neural network (SO-PIFRNN) framework, which significantly improves the numerical solving accuracy of PDEs through hyperparameter optimization mechanism. The framework employs a bi-level optimization architecture: the outer-level optimization utilizes a multi-strategy collaborated particle swarm optimization (MSC-PSO) algorithm to search for optimal hyperparameters of physics-informed Fourier-features randomized neural network, while the inner-level optimization determines the output layer weights of the neural network via the least squares method. The core innovation of this study is embodied in the following three aspects: First, the Fourier basis function activation mechanism is introduced in the hidden layer of neural network, which significantly enhances the ability of the network to capture multi-frequency components of the solution. Secondly, a novel derivative neural network method is proposed, which improves the calculation accuracy and efficiency of PIFRNN method. Finally, the MSC-PSO algorithm of the hybrid optimization strategy is designed to improve the global search ability and convergence accuracy through the synergistic effect of dynamic parameter adjustment, elitist and mutation strategies. Through a series of numerical experiments, including multiscale equations in complex regions, high-order equations, high-dimensional equations and nonlinear equations, the validity of SO-PIFRNN is verified. The experimental results affirm that SO-PIFRNN exhibits superior approximation accuracy and frequency capture capability.

cs.NE↗

arXiv2025

Higher-order multiscale method and its convergence analysis for nonlinear thermo-electric coupling problems of composite structures

This paper proposes a higher-order multiscale computational method for nonlinear thermo-electric coupling problems of composite structures, which possess temperature-dependent material properties and nonlinear Joule heating. The innovative contributions of this work are the novel multiscale formulation with the higher-order correction terms for periodic composite structures and the global error estimation with an explicit rate for higher-order multiscale solutions. By employing the multiscale asymptotic approach and the Taylor series technique, the higher-order multiscale method is established for time-dependent nonlinear thermo-electric coupling problems, which can keep the local balance of heat flux and electric charge for high-accuracy multiscale simulation. Furthermore, an efficient numerical algorithm with off-line and on-line stages is presented in detail, and corresponding convergent analysis is also obtained. Two- and three-dimensional numerical experiments are conducted to showcase the competitive advantages of the proposed method for simulating the time-dependent nonlinear thermo-electric coupling problems in composite structures, not only exceptional numerical accuracy, but also less computational cost.

math.NA↗

arXiv2023

Higher-order multi-scale method for high-accuracy nonlinear thermo-mechanical simulation of heterogeneous shells

In the present work, we consider multi-scale computation and convergence for nonlinear time-dependent thermo-mechanical equations of inhomogeneous shells possessing temperature-dependent material properties and orthogonal periodic configurations. The first contribution is that a novel higher-order macro-micro coupled computational model is rigorously devised via multi-scale asymptotic technique and Taylor series approach for high-accuracy simulation of heterogeneous shells. Benefitting from the higher-order corrected terms, the higher-order multi-scale computational model keeps the conservation of local energy and momentum for nonlinear thermo-mechanical simulation. Moreover, a global error estimation with explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. Furthermore, an efficient space-time numerical algorithm with off-line and on-line stages is presented in detail. Adequate numerical experiments are conducted to confirm the competitive advantages of the presented multi-scale approach, exhibiting not only the exceptional numerical accuracy, but also the less computational expense for heterogeneous shells.

math.NA↗

arXiv2023

Higher-order multi-scale deep Ritz method for multi-scale problems of authentic composite materials

The direct deep learning simulation for multi-scale problems remains a challenging issue. In this work, a novel higher-order multi-scale deep Ritz method (HOMS-DRM) is developed for thermal transfer equation of authentic composite materials with highly oscillatory and discontinuous coefficients. In this novel HOMS-DRM, higher-order multi-scale analysis and modeling are first employed to overcome limitations of prohibitive computation and Frequency Principle when direct deep learning simulation. Then, improved deep Ritz method are designed to high-accuracy and mesh-free simulation for macroscopic homogenized equation without multi-scale property and microscopic lower-order and higher-order cell problems with highly discontinuous coefficients. Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorously demonstrated under appropriate assumptions. Finally, extensive numerical experiments are presented to show the computational accuracy of the proposed HOMS-DRM. This study offers a robust and high-accuracy multi-scale deep learning framework that enables the effective simulation and analysis of multi-scale problems of authentic composite materials.

math.NA↗

arXiv2023

Self-optimization wavelet-learning method for predicting nonlinear thermal conductivity of highly heterogeneous materials with randomly hierarchical configurations

In the present work, we propose a self-optimization wavelet-learning method (SO-W-LM) with high accuracy and efficiency to compute the equivalent nonlinear thermal conductivity of highly heterogeneous materials with randomly hierarchical configurations. The randomly structural heterogeneity, temperature-dependent nonlinearity and material property uncertainty of heterogeneous materials are considered within the proposed self-optimization wavelet-learning framework. Firstly, meso- and micro-structural modeling of random heterogeneous materials are achieved by the proposed computer representation method, whose simulated hierarchical configurations have relatively high volume ratio of material inclusions. Moreover, temperature-dependent nonlinearity and material property uncertainties of random heterogeneous materials are modeled by a polynomial nonlinear model and Weibull probabilistic model, which can closely resemble actual material properties of heterogeneous materials. Secondly, an innovative stochastic three-scale homogenized method (STSHM) is developed to compute the macroscopic nonlinear thermal conductivity of random heterogeneous materials. Background meshing and filling techniques are devised to extract geometry and material features of random heterogeneous materials for establishing material databases. Thirdly, high-dimensional and highly nonlinear material features of material databases are preprocessed and reduced by wavelet decomposition technique. The neural networks are further employed to excavate the predictive models from dimension-reduced low-dimensional data.

physics.comp-ph↗

arXiv2022

Convergence analysis of Jacobi spectral collocation methods for weakly singular nonlocal diffusion equations with volume constraints

This paper considers efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. The equation we consider contains an integral operator that typically has a singularity at the midpoint of the integral domain, and the approximation of the integral operator is one of the essential difficulties in solving nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for nonlocal diffusion equations. A rigorous convergence analysis of the proposed method with the $L^\infty$ norm is presented, and we further prove that the Jacobi collocation solution converges to its corresponding local limit as nonlocal interactions vanish. Numerical examples are given to verify the theoretical results.

math.NA↗

arXiv2021

An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains

In this article, an advanced differential quadrature (DQ) approach is proposed for the high-dimensional multi-term time-space-fractional partial differential equations (TSFPDEs) on convex domains. Firstly, a family of high-order difference schemes is introduced to discretize the time-fractional derivative and a semi-discrete scheme for the considered problems is presented. We strictly prove its unconditional stability and error estimate. Further, we derive a class of DQ formulas to evaluate the fractional derivatives, which employs radial basis functions (RBFs) as test functions. Using these DQ formulas in spatial discretization, a fully discrete DQ scheme is then proposed. Our approach provides a flexible and high accurate alternative to solve the high-dimensional multi-term TSFPDEs on convex domains and its actual performance is illustrated by contrast to the other methods available in the open literature. The numerical results confirm the theoretical analysis and the capability of our proposed method finally.

math.NA↗

arXiv2019

Finite Element Methods for Fractional PDEs in Three Dimensions

This paper is a generalization of the previous work (Yang et.al, J. Comput. Phys. 330 (2017), 863-883) to the 3-D irregular convex domains. The analytical calculation formula of fractional derivatives of finite element basis functions are given and a path searching method is developed to find the integration paths corresponding to the Gaussian points. Moreover, a template matrix is introduced to speed up the procedures. Numerical experiments on a steady problem are presented verifying the efficiency of proposed techniques.

math.NA↗

arXiv2019

Provably size-guaranteed mesh generation with superconvergence

The properties and applications of superconvergence on size-guaranteed Delaunay triangulation generated by bubble placement method (BPM), are studied in this paper. First, we derive a mesh condition that the difference between the actual side length and the desired length $h$ is as small as ${\cal O}(h^{1+α})$ $(α>0)$. Second, the superconvergence estimations are analyzed on linear and quadratic finite element for elliptic boundary value problem based on the above mesh condition. In particular, the mesh condition is suitable for many known superconvergence estimations of different equations. Numerical tests are provided to verify the theoretical findings and to exhibit the superconvergence property on BPM-based grids.

math.NA↗

arXiv2018

Superconvergence of Numerical Gradient for Weak Galerkin Finite Element Methods on Nonuniform Cartesian Partitions in Three Dimensions

A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the loss of the symmetric property from two dimensions to three dimensions, this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions \cite{sup_LWW2018} from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the $L^{2}$-norm arrives at a superconvergence order of ${\cal O}(h^r) (1.5 \leq r\leq 2)$ when the lowest order weak Galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed. A series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions.

math.NA↗

arXiv2018

Effective numerical treatment of sub-diffusion equation with non-smooth solution

In this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a smooth operator, we devise a high-order numerical scheme by combining the Nystrom method in temporal direction with the compact finite difference method and the spectral method in spatial direction. The distinct advantage of this approach in comparison with most current methods is its high convergence rate even though the solution of the anomalous sub-diffusion equation usually has lower regularity on the starting point. The effectiveness and efficiency of our proposed method are verified by several numerical experiments.

math.NA↗

arXiv2017

Multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains

This study develops a novel multiscale computational method for heat conduction problems of composite structures with diverse periodic configurations in different subdomains. Firstly, the second-order two-scale (SOTS) solutions for these multiscale problems are successfully obtained based on asymptotic homogenization method. Then, the error analysis in the pointwise sense is given to illustrate the importance of developing SOTS solutions. Furthermore, the error estimates for the SOTS approximate solutions in the integral sense is presented. In addition, a SOTS numerical algorithm is proposed to effectively solve these problems based on finite element method. Finally, some numerical examples verify the feasibility and effectiveness of the SOTS numerical algorithm we proposed.

math.NA↗

arXiv2017

Numerical algorithm for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term

In this paper, an alternating direction implicit (ADI) difference scheme for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term is presented. The unique solvability of the difference solution is discussed, and the unconditional stability and convergence order of the numerical scheme are analysed. Finally, numerical experiments are carried out to verify the effectiveness and accuracy of the algorithm.

math.NA↗

arXiv2016

On a collocation method for the time-fractional convection-diffusion equation with variable coefficients

In this work, a new collocation approach using a combination of a wavelet operational matrix method and the exponential spline interpolation is proposed to solve the time-fractional convection-diffusion equation with variable coefficients. The operational matrix of fractional order integration is first derived based on sine-cosine wavelet functions, which helps to convert the underlying equation into a linear algebraic system. Then, an exponential B-spline method is introduced in spatial direction. On selecting a set of proper collocation points, the method in presence is evaluated on several test problems and the numerical results finally illustrate its validity and applicability.

math.NA↗

arXiv2016

Maximum-norm error analysis of compact difference schemes for the backward fractional Feynman-Kac equation

The fractional Feynman-Kac equations describe the distribution of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the discretized schemes for fractional substantial derivatives proposed recently, we construct compact finite difference schemes for the backward fractional Feynman-Kac equation, which has q-th (q=1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. In the case q=1, the numerical stability and convergence of the difference scheme in the discrete L-infinity norm are proved strictly, where a new inner product is defined for the theoretical analysis. Finally, numerical examples are provided to verify the effectiveness and accuracy of the algorithms.

math.NA↗