Table of Content

20 June 2025, Volume 7 Issue 2

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Preface

Zhong-Zhi Bai, Wei Cai, Qiang Du, Weinan E, Chi-Wang Shu, Xuejun Xu, Zhimin Zhang

2025, 7(2):  409-410.  doi:10.1007/s42967-024-00471-1

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Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes

Xiu Ye, Shangyou Zhang

2025, 7(2):  411-425.  doi:10.1007/s42967-023-00330-5

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A novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation on rectangular meshes. This CDG method with discontinuous P_k (k ≥ 1) elements converges to the true solution two orders above the continuous finite element counterpart. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and the L² norm. A local post-process is defined which lifts a P_k CDG solution to a discontinuous P_k+2 solution. It is proved that the lifted P_k+2 solution converges at the optimal order. The numerical tests illustrate the theoretic findings.

Conforming P₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes

Shangyou Zhang

2025, 7(2):  426-441.  doi:10.1007/s42967-023-00335-0

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The continuous P₃ and discontinuous P₂ finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations. By putting two diagonal lines into every quadrilateral of a quadrilateral mesh, we get a subquadrilateral triangular mesh. Such a velocity solution is divergence-free point wise and viscosity robust in the sense the solution and the error are independent of the viscosity. Numerical examples show an advantage of such a method over the Taylor-Hood P₃-P₂ method, where the latter deteriorates when the viscosity becomes small.

A Partially Random Trotter Algorithm for Quantum Hamiltonian Simulations

Shi Jin, Xiantao Li

2025, 7(2):  442-469.  doi:10.1007/s42967-023-00336-z

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Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising, respectively, from the random sampling of the Hamiltonian terms and the operator-splitting error. Leveraging on the bias/variance tradeoff, the error can be minimized by balancing the two. The concentration bound provides an estimate of the number of gates. The estimates are verified using numerical experiments on classical computers.

A Locking-Free and Reduction-Free Conforming Finite Element Method for the Reissner-Mindlin Plate on Rectangular Meshes

Shangyou Zhang, Zhimin Zhang

2025, 7(2):  470-484.  doi:10.1007/s42967-023-00343-0

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A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation. The rotation is approximated by C¹ - Q_k+1 in one direction and C⁰ - Q_k in the other direction finite elements. The displacement is approximated by C¹ - Q_k+1,k+1. The method is locking-free without using any projection/reduction operator. Theoretical proof and numerical confirmation are presented.

Error Estimates of Finite Element Method for the Incompressible Ferrohydrodynamics Equations

Shipeng Mao, Jiaao Sun

2025, 7(2):  485-535.  doi:10.1007/s42967-023-00347-w

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In this paper, we consider the Shliomis ferrofluid model and study its numerical approximation. We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics (SFHD) equations. First, we establish the wellposedness and some regularity results for the solution of the SFHD model. Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the L²-H¹ error estimates for the time-discrete solution. Moreover, certain regularity results for the time-discrete solution are proved rigorously. With the help of these regularity results, we prove the unconditional L²-H¹ error estimates for the finite element solution of the SFHD model. Finally, some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.

Hierarchical Interpolative Factorization for Self Green’s Function in 3D Modified Poisson-Boltzmann Equations

Yihui Tu, Zhenli Xu, Haizhao Yang

2025, 7(2):  536-561.  doi:10.1007/s42967-023-00352-z

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The modified Poisson-Boltzmann (MPB) equations are often used to describe the equilibrium particle distribution of ionic systems. In this paper, we propose a fast algorithm to solve the MPB equations with the self Green’s function as the self-energy in three dimensions, where the solution of the self Green’s function poses a computational bottleneck due to the requirement of solving a high-dimensional partial differential equation. Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green’s function, building upon our previous result of two dimensions. This approach yields an algorithm with a complexity of O(N log N) by strategically leveraging the locality and low-rank characteristics of the corresponding operators. Additionally, the theoretical O(N) complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction. Extensive numerical results are conducted to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.

A Low-Rank Global Krylov Squared Smith Method for Solving Large-Scale Stein Matrix Equation

Song Nie, Hua Dai

2025, 7(2):  562-588.  doi:10.1007/s42967-023-00364-9

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This paper deals with the numerical solution of the large-scale Stein and discrete-time Lyapunov matrix equations. Based on the global Arnoldi process and the squared Smith iteration, we propose a low-rank global Krylov squared Smith method for solving largescale Stein and discrete-time Lyapunov matrix equations, and estimate the upper bound of the error and the residual of the approximate solutions for the matrix equations. Moreover, we discuss the restarting of the low-rank global Krylov squared Smith method and provide some numerical experiments to show the efficiency of the proposed method.

Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian

Jinye Shen, Bowen Shi, Weizhang Huang

2025, 7(2):  589-605.  doi:10.1007/s42967-024-00368-z

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A so-called grid-overlay finite difference method (GoFD) was proposed recently for the numerical solution of homogeneous Dirichlet boundary value problems (BVPs) of the fractional Laplacian on arbitrary bounded domains. It was shown to have advantages of both finite difference (FD) and finite element methods, including their efficient implementation through the fast Fourier transform (FFT) and the ability to work for complex domains and with mesh adaptation. The purpose of this work is to study GoFD in a meshfree setting, a key to which is to construct the data transfer matrix from a given point cloud to a uniform grid. Two approaches are proposed, one based on the moving least squares fitting and the other based on the Delaunay triangulation and piecewise linear interpolation. Numerical results obtained for examples with convex and concave domains and various types of point clouds are presented. They show that both approaches lead to comparable results. Moreover, the resulting meshfree GoFD converges in a similar order as GoFD with unstructured meshes and finite element approximation as the number of points in the cloud increases. Furthermore, numerical results show that the method is robust to random perturbations in the location of the points.

A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems

Qigang Liang, Wei Wang, Xuejun Xu

2025, 7(2):  606-636.  doi:10.1007/s42967-024-00372-3

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Since the nonconforming finite elements (NFEs) play a significant role in approximating PDE eigenvalues from below, this paper develops a new and parallel two-level preconditioned Jacobi-Davidson (PJD) method for solving the large scale discrete eigenvalue problems resulting from NFE discretization of 2mth (m = 1, 2) order elliptic eigenvalue problems. Combining a spectral projection on the coarse space and an overlapping domain decomposition (DD), a parallel preconditioned system can be solved in each iteration. A rigorous analysis reveals that the convergence rate of our two-level PJD method is optimal and scalable. Numerical results supporting our theory are given.

A Note on Stability Analysis of Two-Dimensional Runge-Kutta Discontinuous Galerkin Methods

Yuan Xu, Qiang Zhang

2025, 7(2):  637-662.  doi:10.1007/s42967-024-00370-5

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In this paper, we shall carry out the L²-norm stability analysis of the Runge-Kutta discontinuous Galerkin (RKDG) methods on rectangle meshes when solving a linear constantcoefficient hyperbolic equation. The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis. This extension looks easy for most cases; however, there are a few troubles with obtaining good stability results under a standard CFL condition, especially, for those Q^k-elements with lower degree k as stated in the one-dimensional case. To overcome this difficulty, we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose. In addition, an optimal error estimate on Q^k-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau (GGR) projection.

An Augmented Two-Scale Finite Element Method for Eigenvalue Problems

Xiaoying Dai, Yunyun Du, Fang Liu, Aihui Zhou

2025, 7(2):  663-688.  doi:10.1007/s42967-024-00375-0

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In this paper, an augmented two-scale finite element method is proposed for a class of linear and nonlinear eigenvalue problems on tensor-product domains. Through a correction step, the augmented two-scale finite element solution is obtained by solving an eigenvalue problem on a low-dimensional augmented subspace. Theoretical analysis and numerical experiments show that the augmented two-scale finite element solution achieves the same order of accuracy as the standard finite element solution on a fine grid, but the computational cost required by the former solution is much lower than that demanded by the latter. The augmented two-scale finite element method also improves the approximation accuracy of eigenfunctions in the L²(Ω) norm compared with the two-scale finite element method.

A Corrected H_N^T-UGKS for Boundary Layer Problems of the Gray Radiative Transfer Equations

Song Jiang, Qi Li, Wenjun Sun

2025, 7(2):  689-717.  doi:10.1007/s42967-024-00376-z

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In this paper, the corrected method to the original H_N^T-unified gas kinetic scheme (H_N^T-UGKS) is developed in order to solve the nonlinear radiative transfer equations with boundary layers. The H_N^T-UGKS is an asymptotic preserving (AP) scheme that uses UGKS for spatial discretization and the hybrid H_N^T method for angular discretization which is constructed in the paper (Li et al. in Nucl. Sci. Eng. 198(5): 993–1020, 2024). First, the correction idea in Mieussens (J. Comput. Phys. 253: 138–156, 2013) is adopted, such that H_N^T-UGKS can correctly simulate the linear radiative transfer equation with boundary layers. Then, for the nonlinear radiative transfer equations with boundary layers, the transformation from the implicit Monte Carlo (IMC) method is introduced to rewrite the nonlinear transfer equations into a linearized system. It is the key point in the construction of the current scheme to use this linearized system to construct the numerical boundary fluxes. In this way, the boundary density is included in the numerical fluxes, and consequently, the modification method for the linear radiative transfer equation can be used to deal with the nonlinear problem studied in this paper. A number of numerical examples are presented to demonstrate the accuracy and effectiveness of the current scheme for resolving boundary layers in both linear and nonlinear radiative transfer problems.

Immunity-Based Orthogonal Weights Modification Algorithm

Xiaotao Liang, Shuo Yin, Xing Zhao, XuYing Zhao

2025, 7(2):  718-732.  doi:10.1007/s42967-024-00377-y

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Recently, the catastrophic forgetting problem of neural networks in the process of continual learning (CL) has attracted more and more attention with the development of deep learning. The orthogonal weight modification (OWM) algorithm to some extent overcomes the catastrophic forgetting problem in CL. It is well-known that the mapping rule learned by the network is usually not accurate in the early stage of neural network training. Our main idea is to establish an immune mechanism in CL, which rejects unreliable mapping rules at the beginning of the training until those are reliable enough. Our algorithm showed a very good competitive advantage in the permuted and disjoint MNIST tasks and disjoint CIFAR-10 tasks. As for the more challenging task of Chinese handwriting character recognition, our algorithm showed a notable improvement compared with the OWM algorithm. In view of the context-dependent processing (CDP) module in [37], we revealed that the module may result in a loss of information and we proposed a modified CDP module to overcome this weakness. The performance of the system with the modified CDP module outperforms the original one in the CelebFaces attributed recognition task. Besides continual multi-task, we also considered a single task, where the immunity-based OWM (IOWM) algorithm was designed as an optimization solver of neural networks for low-dose computed tomography (CT) denoising task.

Improvement of Convergence of One- and Two-Step MSM Iteration Methods for Nondifferentiable Nonlinear Complementarity Problems

Gui-Lin Yan, Yu-Jiang Wu, Bo Deng

2025, 7(2):  733-758.  doi:10.1007/s42967-024-00378-x

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Our goal is to improve the convergence theory of the one-step modulus-based synchronous multisplitting (MSM) and the two-step modulus-based synchronous multisplitting (TMSM) iteration methods for a class of nondifferentiable nonlinear complementarity problems (NCPs) with H₊-matrices. The analysis is developed and the results are renewed under some conditions weakened than before.

On the Regularity of Time-Harmonic Maxwell Equations with Impedance Boundary Conditions

Zhiming Chen

2025, 7(2):  759-770.  doi:10.1007/s42967-024-00386-x

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In this paper, we prove the H² regularity of the solution to the time-harmonic Maxwell equations with impedance boundary conditions on domains with a C² boundary under minimum regularity assumptions on the source and boundary functions.

Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

Mengfei Wang, Yan Xu

2025, 7(2):  771-795.  doi:10.1007/s42967-024-00390-1

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This paper concerns the superconvergence property of the ultraweak-local discontinuous Galerkin (UWLDG) method for one-dimensional linear sixth-order equations. The crucial technique is the construction of a special projection. We will discuss in three different situations according to the remainder of k, the highest degree of polynomials in the function space, divided by 3. We can prove the (2k - 1) th-order superconvergence for the cell averages when k ≡ 0 or 2 (mod 3). But if k ≡1 (mod 3), we can only prove a (2k - 2) th-order superconvergence. The same superconvergence orders can also be gained for the errors of numerical fluxes. We will also prove the superconvergence of order k + 2 at some special quadrature points. Some numerical examples are given at the end of this paper.

Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Method for 1D Conservation Laws

Lei Yang, Shun Li, Yan Jiang, Chi-Wang Shu, Mengping Zhang

2025, 7(2):  796-826.  doi:10.1007/s42967-024-00391-0

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In this paper, we propose a new class of discontinuous Galerkin (DG) methods for solving 1D conservation laws on unfitted meshes. The standard DG method is used in the interior cells. For the small cut elements around the boundaries, we directly design approximation polynomials based on inverse Lax-Wendroff (ILW) principles for the inflow boundary conditions and introduce the post-processing to preserve the local conservation properties of the DG method. The theoretical analysis shows that our proposed methods have the same stability and numerical accuracy as the standard DG method in the inner region. An additional nonlinear limiter is designed to prevent spurious oscillations if a shock is near the boundary. Numerical results indicate that our methods achieve optimal numerical accuracy for smooth problems and do not introduce additional oscillations in discontinuous problems.