14 August 2025, Volume 47 Issue 3

    

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Youth Review
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  INTRINSIC FINITE ELEMENTS: CURRENTS, DISCRETE GEOMETRY AND PERIODIC TABLE

  Hu Kaibo

  Mathematica Numerica Sinica. 2025, 47(3): 385-417. https://doi.org/10.12286/jssx.j2025-1308

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  This paper focuses on intrinsic finite elements, exploring their applications in numerical partial differential equations and their potential connections to discrete differential geometry and topological data analysis. Driven by the numerical discretization that preserves the mathematical and physical structures of continuous problems, the paper briefly reviews the development of Finite Element Exterior Calculus (FEEC). Through the canonical discretization of the classical de Rham complex and the BGG complex, an extended finite element periodic table for form-valued differential forms is proposed, covering Whitney forms, distributional finite elements, Regge finite elements, and Hessian and div div complexes, providing a unified tool for the numerical solution of tensor problems. The paper further analyzes the potential of intrinsic finite elements in interdisciplinary applications, including Riemann-Cartan geometry, generalized continua, and gravitational wave computations.
Articles
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  LINEAR FIRST-ORDER NUMERICAL SCHEMES FOR A NONLINEAR WAVE EQUATION BASED ON EXPONENTIAL SCALAR AUXILIARY VARIABLE

  Li Huanhuan, Li Meng Luo, Xianbing

  Mathematica Numerica Sinica. 2025, 47(3): 418-435. https://doi.org/10.12286/jssx.j2024-1235

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  In recent years, the exponential scalar auxiliary variable (E-SAV) method is very popular for approximating the phase field models. This method is very effective and does not have to assume that the nonlinear function is bounded from below. In the current study, an E-SAV method is proposed for the numerical investigation of a nonlinear wave equation. This scheme with first order accuracy is obtained by using two variables and backward Euler formula. The error of the approximation of the proposed scheme for the nonlinear wave equation is analyzed. To verify the theoretical results, and the effectiveness of the method with other important methods, two numerical experiments are carried out.
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  AN ACCELERATED PRIMAL-DUAL FIXED POINT ALGORITHM FOR THREE-BLOCK COMPOSITE OPTIMIZATION PROBLEMS

  Luo Yueying, Cai Xingju, Sun Yuehong

  Mathematica Numerica Sinica. 2025, 47(3): 436-450. https://doi.org/10.12286/jssx.j2024-1236

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  For the composite optimization problems widely encountered in machine learning and image processing, the Primal-Dual Fixed Point(PDFP) algorithm is an efficient algorithm. In this paper, we propose an Accelerated Primal-Dual Fixed Point algorithm(APDFP) by combining the PDFP and Nesterov acceleration technique. APDFP can encompass the Accelerated Proximal Alternating Prediction Corrector Algorithm(APAPC) as a special case. Under appropriate conditions, we prove that APDFP has a non-ergodic convergence rate of O(1/N). Furthermore, numerical experiments on the fused lasso problem and computed tomography(CT) image reconstruction verify the effectiveness of the proposed algorithm.
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  IMPLICIT SEMI-TAMED EULER METHOD FOR HIGHLY NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

  Fan Zhencheng

  Mathematica Numerica Sinica. 2025, 47(3): 451-470. https://doi.org/10.12286/jssx.j2024-1238

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  So far, all of the tamed methods are implicit for highly nonlinear stochastic differential equations. For the stochastic differential equation that its drift coefficients can be divided into a linear term and a highly nonlinear term, we present the implicit semi-tamed Euler methods which only require similar computational effort as an explicit method. Under the Khasminskii-type and polynomial growth conditions, the convergence of the presented method is proved. In addition, we study the stability of the method and prove that it can preserve the stability of analysis solutions of an stable system. Finally, we give some numerical examples to verify the theoretical results and show that the stability of the presented methods is better than that of some explicit tamed methods.
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  OPTIMAL ERROR ESTIMATES OF NONCONFORMING VIRTUAL ELEMENT METHOD FOR NONLINEAR SOBOLEV EQUATIONS ON POLYGONAL MESHES

  Zhu Peng, Chen Yanping, Liu Wanxiang

  Mathematica Numerica Sinica. 2025, 47(3): 471-489. https://doi.org/10.12286/jssx.j2024-1244

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  In this paper, we propose a nonconforming virtual element method for nonlinear Sobolev equation on polygonal meshes by applying backward Euler scheme. In order to establish the convergence of the method, we construct a novel projection operator based on two discrete trilinear forms and give the corresponding error estimates in the L² norm and broken H¹ semi-norm. Leveraging this projection operator, we prove the optimal convergence of the nonconforming virtual element method in the fully discrete formulation. Finally, several numerical experiments on various polygonal meshes are conducted to confirm the accuracy and optimal convergence of the proposed method.
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  AN ACCELERATED SURROGATE HYPERPLANE KACZMARZ METHOD BASED ON TWO-DIMENSIONAL SEARCH

  Guo Xuan, Li Rui, Yin Junfeng

  Mathematica Numerica Sinica. 2025, 47(3): 490-501. https://doi.org/10.12286/jssx.j2024-1247

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  An accelerated surrogate hyperplane Kaczmarz method based on two-dimensional search is proposed, which generates a new hyperplane by searching for the optimal weighted vector in a two-dimensional subspace. Theoretical analysis provides the convergence rate of the new method. Numerical experiments demonstrate that the new proposed Kaczmarz method is convergent and outperforms the original method in terms of both the number of iterations and computational time.
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  CONSTRAINED RANDOMIZED GAUSS-SEIDEL METHODS FOR NONNEGATIVE LEAST-SQUARES PROBLEM

  Li Xiaoling, Wei Wei, Shi Tao

  Mathematica Numerica Sinica. 2025, 47(3): 502-518. https://doi.org/10.12286/jssx.j2024-1249

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  For solving the large-scale overdetermined linear least-squares problem with nonnegative constraints, we propose two constrained Gauss-Seidel methods, namely constrained greedy randomized Gauss-Seidel method based on greedy criterion and constrained random sampling Gauss-Seidel method based on random sampling. We also build the convergence theories and implement some numerical experiments in this paper. The numerical results demonstrate that the proposed methods significantly outperform the existing methods.
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  LAGRANGE MULTIPLIER METHOD FOR CAHN-HILLIARD-HELE-SHAW MODEL

  Liu Jianghua, Zhai Shuying, Li Xiaoli

  Mathematica Numerica Sinica. 2025, 47(3): 519-534. https://doi.org/10.12286/jssx.j2024-1252

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  The Cahn-Hilliard-Hele-Shaw (CHHS) model, a Cahn-Hilliard equation coupled with the Darcy equation. It has been widely used to simulate two-phase flow in porous media and tumor growth. For the CHHS model, two energy dissipation schemes based on the Lagrange multiplier method are proposed in this paper. The Backward-Euler and Crank-Nicolson schemes are used in the time direction, and the Fourier spectral method is used in the space direction. Theoretical analysis shows that resulted schemes maintain the original energy dissipation. Finally, various numerical simulations are performed to validate the accuracy and efficiency of the proposed schemes.
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  A TWO-LEVEL ALGORITHM OF THE FINITE ELEMENT METHOD FOR A CLASS OF TIME-DEPENDENT PNP EQUATIONS

  Mao Wantao, Shen Ruigang, Yang Ying

  Mathematica Numerica Sinica. 2025, 47(3): 535-546. https://doi.org/10.12286/jssx.j2024-1254

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  A two-level algorithm based on the finite element discretization is proposed for a class of time-dependent Poisson-Nernst-Planck (PNP) equations. This algorithm decouples the PNP equations through a linear finite element approximation, followed by solving the decoupled equations in a quadratic finite element space. Compared with the classical Gummel algorithm based on the finite element discretization, this method accelerates the solution process. The H¹ norm error estimates are established based on the L² norm error estimates of the twolevel finite element solution. Numerical experiment confirms the correctness of the theoretical results and shows the efficiency of the two-level algorithm.
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  ARBITRARY HIGH-ORDER ENERGY-PRESERVING SCHEMES FOR THE KLEIN-GORDON EQUATION

  Shu Siqi, Wang Jialing

  Mathematica Numerica Sinica. 2025, 47(3): 547-560. https://doi.org/10.12286/jssx.j2024-1257

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  In this paper, a class of arbitrary high-order energy-preserving schemes for the KleinGordon equation is developed. By introducing the quadratic auxiliary variable, the Hamiltonian energy is transformed into the quadratic form, i.e., the energy conservation law is transformed into the quadratic invariants. Then, the original system is reformulated into a new system with quadratic invariants simultaneously. The fully discrete scheme is derived using the Fourier pseudo-spectral method and the symplectic Runge-Kutta methods. The proposed schemes achieve arbitrarily high-order convergence in time, spectral accuracy in space, and can preserve the original energy conservation precisely. The numerical results further substantiate the effectiveness and high-precision convergence of the proposed schemes.