Hehu Xie

Qun Lin, Hehu Xie, Jinchao Xu

Assume that $V_h$ is a space of piecewise polynomials of degree less than $r\geq 1$ on a family of quasi-uniform triangulation of size $h$. Then the following well-known upper bound holds for a sufficiently smooth function $u$ and $p\in [1, \infty]$ $$ \inf_{v_h\in V_h}\|u-v_h\|_{j,p,Ω,h} \le C h^{r-j} |u|_{r,p,Ω},\quad 0\le j\le r. $$ In this paper, we prove that, roughly speaking, if $u\not\in V_h$, the above estimate is sharp. Namely, $$ \inf_{v_h\in V_h}\|u-v_h\|_{j,p,Ω,h} \ge c h^{r-j},\quad 0\le j\le r, \ \ 1\leq p\leq \infty, $$ for some $c>0$. The above result is further extended to various situations including more general Sobolev space norms, general shape regular grids and many different types of finite element spaces. As an application, the sharpness of finite element approximation of elliptic problems and the corresponding eigenvalue problems is established.

Qun Lin, Hehu Xie

In this paper, a new type of multi-level correction scheme is proposed for solving eigenvalue problems by finite element method. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving eigenvalue problems by finite element method.

Fusheng Luo, Qun Lin, Hehu Xie

This article is devoted to computing the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extension $Q_1^{\rm rot}$, we get the lower bound of the eigenvalue. Additionally, we also use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue. The postprocessing method need only to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once. Some numerical results are also presented to validate our theoretical analysis.

Chun'guang You, Hehu Xie, Xuefeng Liu

To provide mathematically rigorous eigenvalue bounds for the Steklov eigenvalue problem, an enhanced version of the eigenvalue estimation algorithm developed by the third author is proposed, which removes the requirements of the positive definiteness of bilinear forms in the formulation of eigenvalue problems. In practical eigenvalue estimation, the Crouzeix--Raviart finite element method (FEM) along with quantitative error estimation is adopted. Numerical experiments for eigenvalue problems defined on a square domain and an L-shaped domain are provided to validate the precision of computed eigenvalue bounds.

Hehu Xie, Xinming Wu

In this paper, we give a numerical analysis for the transmission eigenvalue problem by the finite element method. A type of multilevel correction method is proposed to solve the transmission eigenvalue problem. The multilevel correction method can transform the transmission eigenvalue solving in the finest finite element space to a sequence of linear problems and some transmission eigenvalue solving in a very low dimensional spaces. Since the main computational work is to solve the sequence of linear problems, the multilevel correction method improves the overfull efficiency of the transmission eigenvalue solving. Some numerical examples are provided to validate the theoretical results and the efficiency of the proposed numerical scheme.

Hehu Xie

A type of adaptive finite element method for the eigenvalue problems is proposed based on the multilevel correction scheme. In this method, adaptive finite element method to solve eigenvalue problems involves solving associated boundary value problems on the adaptive partitions and small scale eigenvalue problems on the coarsest partitions. Hence the efficiency of solving eigenvalue problems can be improved to be similar to the adaptive finite element method for the associated boundary value problems. The convergence and optimal complexity is theoretically verified and numerically demonstrated.

Hehu Xie, Manting Xie

In this paper, we propose a computable error estimate of the Gross-Pitaevskii equation for ground state solution of Bose-Einstein condensates by general conforming finite element methods on general meshes. Based on the proposed error estimate, asymptotic lower bounds of the smallest eigenvalue and ground state energy can be obtained. Several numerical examples are presented to validate our theoretical results in this paper.

Hehu Xie, Fei Xu, Ning Zhang

An efficient multigrid method is proposed to compute the ground state solution of Bose-Einstein condensations by the finite element method based on the combination of the multigrid method for nonlinear eigenvalue problem and an efficient implementation for the nonlinear iteration. The proposed numerical method not only has the optimal convergence rate, but also has the asymptotically optimal computational work which is independent from the nonlinearity of the problem. The independence from the nonlinearity means that the asymptotic estimate of the computational work can reach almost the same as that of solving the corresponding linear boundary value problem by the multigrid method. Some numerical experiments are provided to validate the efficiency of the proposed method.

Hehu Xie, Meiling Yue, Ning Zhang

This paper is concerned with the computable error estimates for the eigenvalue problem which is solved by the general conforming finite element methods on the general meshes. Based on the computable error estimate, we can give an asymptotically lower bound of the general eigenvalues. Furthermore, we also give a guaranteed upper bound of the error estimates for the first eigenfunction approximation and a guaranteed lower bound of the first eigenvalue based on computable error estimator. Some numerical examples are presented to validate the theoretical results deduced in this paper.

Xiaobo Yin, Hehu Xie

A uniform strategy to derive metric tensors in two spatial dimension for interpolation errors and their gradients in $L^p$ norm is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in corresponding metric space, with the metric tensor being computed based on a posteriori error estimates in different norms. Numerical results show that the corresponding convergence rates are always optimal.

Hehu Xie, Fei Xu

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.

Xiaobo Yin, Hehu Xie

In this paper, we give a new type of a posteriori error estimators suitable for moving finite element methods under anisotropic meshes for general second-order elliptic problems. The computation of estimators is simple once corresponding Hessian matrix is recovered. Wonderful efficiency indices are shown in numerical experiments.

Yunhui He, Hehu Xie

In this paper, we consider the shift-inverse method with Richardson iteration step for the eigenvalue problems. It will be shown that the convergence speed depends heavily on the eigenvalue gap between the desired eigenvalue and undesired ones.

Hehu Xie, Fei Xu, Meiling Yue

In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose-Einstein condensates based on Newton iteration method. Instead of treating eigenvalue $λ$ and eigenvector $u$ respectively, we regard the eigenpair $(λ, u)$ as one element in the composite space $\R \times H_0^1(Ω)$ and then Newton iteration method is adopted for the nonlinear problem. Thus in this multigrid scheme, we only need to solve a linear discrete boundary value problem in every refined space, which can improve the overall efficiency for the simulation of Bose-Einstein condensations.

Yana Di, Hehu Xie, Xiaobo Yin

We propose a numerical strategy to generate the anisotropic meshes and select the appropriate stabilized parameters simultaneously for two dimensional convection-dominated convection-diffusion equations by stabilized continuous linear finite elements. Since the discretized error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective fields, and the geometric properties such as directed edges and the area of the triangle. Based on this observation, the shape, size and equidistribution requirements are used to derive the corresponding metric tensor and the stabilized parameters. It is easily found from our derivation that the optimal stabilized parameter is coupled with the optimal metric tensor on each element. Some numerical results are also provided to validate the stability and efficiency of the proposed numerical strategy.

Yunhui He, Yu Li, Hehu Xie

We propose a new type of multilevel method for solving eigenvalue problems based on Newton iteration. With the proposed iteration method, solving eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and solving a series of augmented linear problems, derived by Newton step in the corresponding series of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue problems. Finally, some numerical examples are provided to validate the efficiency of the proposed numerical scheme.

Yu Li, Qun Lin, Hehu Xie

In this paper, we present a parallel scheme to solve the population balance equations based on the method of characteristics and the finite element discretization. The application of the method of characteristics transform the higher dimensional population balance equation into a series of lower dimensional convection-diffusion-reaction equations which can be solved in a parallel way.Some numerical results are presented to show the accuracy and efficiency.

Hehu Xie

In this paper, the stabilized finite element method based on local projection is applied to discretize the Stokes eigenvalue problems and the corresponding convergence analysis is given. Furthermore, we also use a method to improve the convergence rate for the eigenpair approximations of the Stokes eigenvalue problem. It is based on a postprocessing strategy that contains solving an additional Stokes source problem on an augmented finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of mixed finite element space. Numerical examples are given to confirm the theoretical analysis.

Xiaobo Yin, Hehu Xie

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on a posteriori error estimates proposed in \cite{YinXie}. The new metric tensor explores more comprehensive information of anisotropy for the true solution than those existing ones. Numerical results show that this approach can be successfully applied to deal with poisson and steady convection-dominated problems. The superior accuracy and efficiency of the new metric tensor to others is illustrated on various numerical examples of complex two-dimensional simulations.

Chenhao Jin, Hehu Xie

This paper develops a robust solver for the Maxwell eigenproblem in 3D photonic crystals with anisotropic media. The solver employs the kernel compensation technique under the framework of Yee's scheme to eliminate null space and enable matrix-free, GPU-accelerated operations via 3D discrete Fourier transform. Furthermore, we propose a novel discretization for permittivity tensor containing off-diagonal entries and prove that the resulting matrix is Hermitian positive definite, which ensures the correctness of the kernel compensation technique. Numerical experiments on several benchmark examples are demonstrated to validate the robustness and accuracy of our scheme.

Jingze Ren, Yifan Wang, Hehu Xie et al.

In this paper, we propose a novel machine learning method based on an adaptive tensor neural network subspace for solving quasiperiodic elliptic problems. To this end, we first provide a theoretical analysis of the associated quasiperiodic and periodic function spaces and establish regularity estimates for the quasiperiodic elliptic problems. In particular, under the Diophantine condition, we derive a suitable condition on the source term to guarantee the regularity of the solution, which provides a theoretical basis for the design of numerical schemes. An efficient numerical method is then designed by combining the projection method with tensor neural networks. Leveraging the special structure of tensor neural networks, high-dimensional integration can be performed directly and with high accuracy, without relying on Monte Carlo methods. Finally, several numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method.

Qingkui Ma, Hehu Xie, Xiaobo Yin

Fractional PDEs involving the fractional Laplacian on bounded domains are challenging because of hypersingular nonlocal kernels, exterior Dirichlet constraints, reduced boundary regularity, and the high computational cost in high dimensions. To address these issues, we first adopt a spatially varying radius with directional distance-to-boundary information, which yields a geometry-adaptive three-part decomposition of the fractional Laplacian: singular near-field, regular interior far-field, and analytical exterior far-field contributions. Then we employ Gauss-Jacobi quadrature for the singular radial integral, Gauss quadrature for the regular interior radial integral, and Monte Carlo sampling for the angular variables. A feature-enhanced physics-informed neural network trial space is finally used to tackle the low-regularity behavior near the boundary. Through the above steps, we obtain a quadrature-enhanced Monte Carlo fractional physics-informed neural network (QE-MC-fPINN) method. Numerical experiments on fractional Poisson equations and time-dependent fractional PDEs show that, on the tested benchmarks, the proposed method outperforms two representative MC-fPINN discretizations in accuracy and convergence, especially for solutions with strong boundary singularities.

Zhongshuo Lin, Qingkui Ma, Hehu Xie et al.

In this paper, we propose a novel machine learning method based on adaptive tensor neural network subspace to solve linear time-fractional diffusion-wave equations and nonlinear time-fractional partial integro-differential equations. In this framework, the tensor neural network and Gauss-Jacobi quadrature are effectively combined to construct a universal numerical scheme for the temporal Caputo derivative with orders spanning $ (0,1)$ and $(1,2)$. Specifically, in order to effectively utilize Gauss-Jacobi quadrature to discretize Caputo derivatives, we design the tensor neural network function multiplied by the function $t^μ$ where the power $μ$ is selected according to the parameters of the equations at hand. Finally, some numerical examples are provided to validate the efficiency and accuracy of the proposed tensor neural network based machine learning method.

Yongxin Li, Yifan Wang, Zhongshuo Lin et al.

This paper introduces a tensor neural network (TNN) to address nonparametric regression problems, leveraging its distinct sub-network structure to effectively facilitate variable separation and enhance the approximation of complex, high-dimensional functions. The TNN demonstrates superior performance compared to conventional Feed-Forward Networks (FFN) and Radial Basis Function Networks (RBN) in terms of both approximation accuracy and generalization capacity, even with a comparable number of parameters. A significant innovation in our approach is the integration of statistical regression and numerical integration within the TNN framework. This allows for efficient computation of high-dimensional integrals associated with the regression function and provides detailed insights into the underlying data structure. Furthermore, we employ gradient and Laplacian analysis on the regression outputs to identify key dimensions influencing the predictions, thereby guiding the design of subsequent experiments. These advancements make TNN a powerful tool for applications requiring precise high-dimensional data analysis and predictive modeling.

Qilong Zhai, Hehu Xie, Ran Zhang et al.

Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds property of the weak Galerkin method. Some numerical examples are provided to validate our theoretical analysis.

Yunhui He, Qichen Hong, Hehu Xie et al.

This paper is to give a new understanding and applications of the subspace projection method for selfadjoint eigenvalue problems. A new error estimate in the energy norm, which is induced by the stiff matrix, of the subspace projection method for eigenvalue problems is given. The relation between error estimates in $L^2$-norm and energy norm is also deduced. Based on this relation, a new type of inverse power method is designed for eigenvalue problems and the corresponding convergence analysis is also provided. Then we present the analysis of the geometric and algebraic multigrid methods for eigenvalue problems based on the convergence result of the new inverse power method.

Hehu Xie, Zhimin Zhang

A multilevel correction scheme is proposed to solve defective and nodefective of nonsymmetric partial differential operators by the finite element method. The method includes multi correction steps in a sequence of finite element spaces. In each correction step, we only need to solve two source problems on a finer finite element space and two eigenvalue problems on the coarsest finite element space. The accuracy of the eigenpair approximation is improved after each correction step. This correction scheme improves overall efficiency of the finite element method in solving nonsymmetric eigenvalue problems.

Hehu Xie, Chunguang You

This paper gives a framework to produce the lower bound of eigenvalues defined in a Hilbert space by the eigenvalues defined in another Hilbert space. The method is based on using the max-min principle for the eigenvalue problems.

Hehu Xie

In this paper, we derive an asymptotic error expansion for the eigenvalue approximations by the lowest order Raviart-Thomas mixed finite element method for the general second order elliptic eigenvalue problems. Extrapolation based on such an expansion is applied to improve the accuracy of the eigenvalue approximations. Furthermore, we also prove the superclose property between the finite element projection with the finite element approximation of the eigenvalue problems by mixed finite element methods. In order to prove the full order of the eigenvalue extrapolation, we first propose "the auxiliary equation method". The result of this paper provides a general procedure to produce an asymptotic expansions for eigenvalue approximations by mixed finite elements.