Journal of Computational Mathematics
type: channel
description: Journal of Computational Mathematics (JCM) is an international scientific computing journal founded by Professor Feng Kang in 1983, which is the first Chinese computational mathematics journal published in English. JCM covers all branches of modern computational mathematics such as numerical linear algebra, numerical optimization, computational geometry, numerical PDEs, and inverse problems. JCM has been sponsored by the Institute of Computational Mathematics of the Chinese Academy of Sciences.
publisher: Global Science Press
language: en
publication name: Journal of Computational Mathematics
issn: 0254-9409
items: On the Superconvergence of Collocation Methods Based on Two Post-Processing Techniques for Fractional Integro-Differential Equations with Weakly Singular Kernels
items: Characterizing the Rate of Convergence of the Augmented Lagrange Method for Nonlinear Programming
items: An Accelerated Preconditioned Primal-Dual Gradient Algorithm for Nonconvex Composite Optimization Problems with Applications
items: Finite Element Method with Grünwald-Letnikov Type Approximation in Time for a Constant Time Delay Subdiffusion Equation
items: The Quadratic Discontinuous Finite Volume Element Schemes for Elliptic Problems
items: An Analysis of the Saturation Assumption for Polynomial Interpolation with Application to Adaptive Finite Element Methods
items: A Multiple-Invariants Preserving Scheme for Modified Two-Component Euler-Poincaré Equations
items: High Oder Probabilistic Numerical Methods for Forward Backward Stochastic Differential Equations
items: Regularization Method with Two Differential Operators for Simultaneous Inversion of Source Term and Initial Value in a Time-Fractional Black-Scholes Equation
items: A Structure-Preserving Nonconforming FEM of Nonlinear Kirchhoff-Type Equation with Damping
items: Serendipity Virtual Element Method for the Second Order Elliptic Eigenvalue Problem in Two and Three Dimensions
items: Multi-Level Mixed Finite Element Methods Based on Different Iterations for the Steady Boussinesq Problem
items: A Low Order Globally Divergence-Free WG Finite Element Method for Steady Thermally Coupled Incompressible MHD Flow
items: A Novel TGMFECN Dimension Reduction Method Based on POD for the Improved Nonlinear Fourth-Order Cahn-Hilliard Equations
items: Low-Degree $\mathcal{O}(h^2)$-Accurate Conservative Finite Element Schemes for Incompressible Stokes Problems on General Triangulations
items: Mean-Square Convergence of Two-Step Milstein Methods for Nonlinear Stochastic Delay Differential Equations
items: On the Adaptive Deterministic Block Coordinate Descent Method with Momentum for Solving Large Linear Least-Squares Problems
items: Stochastic Variance Reduced Gradient for Tensor Recovery
items: Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation
items: A Characteristic Mixed Finite Element Two-Grid Method for Compressible Miscible Displacement Problem
items: Primal-Dual Path-Following Methods and the Trust-Region Updating Strategy for Linear Programming with Noisy Data
items: Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data
items: Heavy Ball Flexible GMRES Method for Nonsymmetric Linear Systems
items: Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations
items: A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations
items: Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations
items: A Simple Error Estimate of Discontinuous Galerkin Methods for Elliptic Equations with Low Regularity
items: Inverse Nodal Problem for the Sturm-Liouville Operator with Three Discontinuities
items: A Second-Order in Time and Energy-Dissipative Scheme for Time-Fractional Navier-Stokes Equations
items: Compact Finite Difference Scheme for Some Sobolev Type Equations with Dirichlet Boundary Condition

On the Superconvergence of Collocation Methods Based on Two Post-Processing Techniques for Fractional Integro-Differential Equations with Weakly Singular Kernels
type: item
description: The purpose of this paper is to investigate the superconvergence of collocation methods for fractional integro-differential equations (FIDEs) with weakly singular kernels and Caputo derivative of order $0 < \alpha < 1$. First, the initial value problem of FIDEs is reformulated as a weakly singular Volterra integral equation (VIE), and the existence, uniqueness, and regularity of the exact solution for the original FIDE are obtained with the help of the resolvent theory of VIEs, and it is shown that the singularity of the exact solution is governed by the Caputo derivative, not the weakly singular kernel. Next, the piecewise polynomial collocation method is employed to solve the reformulated VIE numerically, and the optimal convergence order of the collocation solution is obtained on graded meshes. In order to improve the numerical accuracy, two types of postprocessing techniques are used – one is the classical iterated technique for VIEs and another one is the interpolation postprocessing technique. The superconvergence is thoroughly investigated and the optimal superconvergence orders are obtained for both of these two postprocessing techniques. Compared to the classical iterated collocation method, the interpolation postprocessing method has a lower calculation cost. The theoretical results are illustrated by numerical experiments.
subject: Articles
  topic: section
subject: Fractional integro-differential equation
  topic: keywords
subject: Volterra integral equations
  topic: keywords
subject: Collocation methods
  topic: keywords
subject: Interpolation postprocessing
  topic: keywords
subject: Superconvergence
  topic: keywords
creator: Zhenrong Chen
creator: Yanping Chen
creator: Jianwei Zhou
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-03-23
publication date: 2026-03-23
doi: 10.4208/jcm.2510-m2024-0290

Characterizing the Rate of Convergence of the Augmented Lagrange Method for Nonlinear Programming
type: item
description: The rate of convergence of the augmented Lagrangian method for solving nonlinear programming is studied under the Jacobian uniqueness conditions. It is demonstrated that, for a given multiplier vector $(\mu, \lambda)$, the rate of convergence of the augmented Lagrangian method is linear with respect to $\| (\mu, \lambda) - (\mu^{*}, \lambda^{*}) \|$ and the ratio constant is proportional to $1/c$ when the ratio $\| (\mu, \lambda)-(\mu^{*}, \lambda^{*}) \| /c$ is small enough, where $c$ is the penalty parameter that exceeds a threshold $c^{*} > 0$ and $(\mu^{*}, \lambda^{*})$ is the multiplier corresponding to a local minimum point. Importantly, the ratio constant of the $Q$-linear convergence of the sequence of multiplier vectors is estimated by the second-order derivative of the value function of the nonlinear optimization problem. This characterization gives an explicit expression for the rate constant of the $Q$-linear convergence of the sequence of multiplier vectors.
subject: Articles
  topic: section
subject: Nonlinear programming
  topic: keywords
subject: Jacobian uniqueness conditions
  topic: keywords
subject: Augmented Lagrangian method
  topic: keywords
subject: Rate of convergence
  topic: keywords
subject: Value function
  topic: keywords
creator: Yule Zhang
creator: Jihong Zhang
creator: Jia Wu
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-03-23
publication date: 2026-03-23
doi: 10.4208/jcm.2510-m2024-0216

An Accelerated Preconditioned Primal-Dual Gradient Algorithm for Nonconvex Composite Optimization Problems with Applications
type: item
description: In this paper, we consider a class of three-composite nonconvex optimization problems, in which the nonsmooth function is further composed with a linear operator. This problem has many applications such as sparse signal recovery, image processing and machine learning. Based on the conjugate duality theory, we present an accelerated preconditioned primal-dual gradient algorithm for this problem. Compared with the existing algorithms, our algorithm only needs to calculate the proximal mapping of the conjugate function $h^∗$ which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. This may significantly reduce the computation load. We prove that the sequence generated by the proposed algorithm globally converges to a critical point when the function satisfies the Kurdyka- Lojasiewicz property. We also obtain the convergence rate of the proposed algorithm. Finally, numerical results on sparse signal recovery and image processing illustrate the efficiency and competitiveness of the proposed algorithm.
subject: Articles
  topic: section
subject: Nonconvex composite optimization
  topic: keywords
subject: Primal-dual algorithm
  topic: keywords
subject: Convergence
  topic: keywords
subject: Sparse signal recovery
  topic: keywords
subject: Image processing
  topic: keywords
creator: Xian-Jun Long
creator: Jia-Lin Nie
creator: Gao-Xi Li
creator: Zai-Yun Peng
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-03-19
publication date: 2026-03-19
doi: 10.4208/jcm.2512-m2024-0107

Finite Element Method with Grünwald-Letnikov Type Approximation in Time for a Constant Time Delay Subdiffusion Equation
type: item
description: In this work, a subdiffusion equation with constant time delay $\tau$ is considered. First, the regularity of the solution to the considered problem is investigated, finding that its first-order time derivative exhibits singularity at $t = 0^+$ and its second-order time derivative shows singularity at both $t = 0^+$ and $\tau^+$, while the solution can be decomposed into its singular and regular components. Then, we derive a fully discrete finite element scheme to solve the considered problem based on the standard Galerkin finite element method in space and the Grünwald-Letnikov type approximation in time. The analysis shows that the developed numerical scheme is stable. In order to discuss the error estimate, a new discrete Grönwall inequality is established. Under the above decomposition of the solution, we obtain a local error estimate in time for the developed numerical scheme. Finally, some numerical tests are provided to support our theoretical analysis.
subject: Articles
  topic: section
subject: Subdiffusion equation with constant time delay
  topic: keywords
subject: Finite element method
  topic: keywords
subject: Grünwald-Letnikov type approximation
  topic: keywords
subject: Discrete Grönwall inequality
  topic: keywords
subject: Error estimate
  topic: keywords
creator: Weiping Bu
creator: Xueqin Zhang
creator: Weizhi Liao
creator: Yue Zhao
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-03-16
publication date: 2026-03-16
doi: 10.4208/jcm.2510-m2025-0131

The Quadratic Discontinuous Finite Volume Element Schemes for Elliptic Problems
type: item
description: The objective of the paper is to develop the quadratic discontinuous finite volume methods (DFVM) for solving elliptic equations with variable coefficients. The proposed numerical schemes are composed of the discontinuous Galerkin (DG) method with different interior penalty formulations (IIPG, NIPG, SIPG) and the finite volume element method which the trial function is discontinuous quadratic element function. Subsequently, with the specialized projection techniques, we built up a bridge between the bilinear form of DFVM and that of DG method, which simplifies the analysis and proves the optimal error estimate of the schemes in the broken $H^1$ norm. Finally, we provide numerical simulations to validate the theoretical findings.
subject: Articles
  topic: section
subject: Discontinuous finite volume element method
  topic: keywords
subject: Quadratic element
  topic: keywords
subject: Error estimates
  topic: keywords
subject: Elliptic problems
  topic: keywords
creator: Yuzhi Lou
creator: Hongxing Rui
creator: Xiufang Feng
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-03-02
publication date: 2026-03-02
doi: 10.4208/jcm.2509-m2024-0241

An Analysis of the Saturation Assumption for Polynomial Interpolation with Application to Adaptive Finite Element Methods
type: item
description: The saturation assumption plays a central role in much of the analysis of a posteriori error estimates and refinement algorithms for adaptive finite element methods. In this work we provide an analysis of this assumption in the simple setting of interpolation. We have proved elsewhere [Bank and Yserentant, Numer. Math., 131:1 (2015)] that interpolation error is both reliable and efficient as an a posteriori error estimate. Thus behavior of interpolation error is indicative of the behavior of the error in the exact finite element solution of a PDE as well as any practical a posteriori error estimate that is also reliable and efficient.
subject: Articles
  topic: section
subject: Saturation assumption
  topic: keywords
subject: Adaptive feedback loop
  topic: keywords
subject: A posteriori error estimates
  topic: keywords
creator: Randolph E. Bank
creator: Jinchao Xu
creator: Harry Yserentant
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-03-02
publication date: 2026-03-02
starting page: 871
ending page: 890
doi: 10.4208/jcm.2510-m2024-0087

A Multiple-Invariants Preserving Scheme for Modified Two-Component Euler-Poincaré Equations
type: item
description: This paper presents a high-order multiple-invariants preserving numerical scheme for the modified Euler-Poincaré equations with two components. It is shown that the scheme preserves at least three invariants: mass, momentum and energy. In contrast, the previous schemes usually keep only one or two. Meanwhile, the scheme achieves high order accuracy in spatial direction as well as second-order in time, which will be proved rigorously in this paper. The key to the present scheme aims at the construction of a bi-variate function and utilization of a special time discretization. Numerical tests are given to verify the theoretical findings.
subject: Articles
  topic: section
subject: Modified Two-Component Euler-Poincaré Equations
  topic: keywords
subject: Bi-variate function
  topic: keywords
subject: Mass conservation
  topic: keywords
subject: Preserving energy
  topic: keywords
subject: Momentum conservation
  topic: keywords
creator: Boya Zhou
creator: Ruimin Gao
creator: Qifeng Zhang
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-03-02
publication date: 2026-03-02
doi: 10.4208/jcm.2509-m2024-0222

High Oder Probabilistic Numerical Methods for Forward Backward Stochastic Differential Equations
type: item
description: In this paper, we design novel high order probabilistic numerical algorithms for forward backward stochastic differential equations. Moreover, we derive the error estimates and prove the high order convergence rates of the proposed schemes. Because the proposed scheme involves conditional expectations, an estimator based on the multilevel Monte Carlo method is applied to approximate the conditional expectations. Furthermore, we theoretically demonstrate that the computational complexity of our numerical method is proportional to the square of prescribed accuracy. Numerical experiments are given to illustrate the theoretical results.
subject: Articles
  topic: section
subject: Forward backward stochastic differential equation
  topic: keywords
subject: Probabilistic numerical method
  topic: keywords
subject: Multilevel Monte Carlo
  topic: keywords
subject: Computational complexity
  topic: keywords
subject: High order discretization
  topic: keywords
creator: Qiang Han
creator: Yurong Liu
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-02-24
publication date: 2026-02-24
doi: 10.4208/jcm.2509-m2025-0064

Regularization Method with Two Differential Operators for Simultaneous Inversion of Source Term and Initial Value in a Time-Fractional Black-Scholes Equation
type: item
description: This paper is devoted to identifying the source term and initial value simultaneously in a time-fractional Black-Scholes equation, which is an ill-posed problem. The inverse problem is transformed into a system of operator equations, and under certain source conditions, conditional stability is established. We propose a regularization method with two differential operators to solve the problem, error estimates by rules of a priori and a posteriori regularization parameter selection are derived, respectively. Numerical experiments are presented to validate the effectiveness of the proposed regularization method.
subject: Articles
  topic: section
subject: Regularization method with two differential operators
  topic: keywords
subject: Simultaneous inversion of source term and initial value
  topic: keywords
subject: Ill-posed problem
  topic: keywords
subject: Time-fractional Black-Scholes equation
  topic: keywords
creator: Shuang Yu
creator: Hongqi Yang
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-01-28
publication date: 2026-01-28
doi: 10.4208/jcm.2508-m2025-0012

A Structure-Preserving Nonconforming FEM of Nonlinear Kirchhoff-Type Equation with Damping
type: item
description: Superconvergent behavior for nonlinear Kirchhoff-type with damping is researched by a structure-preserving nonconforming finite element method (FEM). A new implicit energy dissipation scheme is developed and the numerical solution is bounded in energy norm. The existence of the numerical solution is obtained with the help of the Brouwer fixed-point theorem and then the uniqueness is gained. Superconvergence characteristics is revealed by the properties of the nonconforming FE and a special splitting technique. Numerical tests confirm the correctness of the theoretical research results.
subject: Articles
  topic: section
subject: A structure-preserving nonconforming finite element method
  topic: keywords
subject: A splitting technique
  topic: keywords
subject: Kirchhoff-type equation with damping
  topic: keywords
subject: Superconvergent behavior
  topic: keywords
creator: Junjun Wang
creator: Dongyang Shi
creator: Shuo Wang
creator: Xuesong Che
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-01-26
publication date: 2026-01-26
doi: 10.4208/jcm.2509-m2024-0004

Serendipity Virtual Element Method for the Second Order Elliptic Eigenvalue Problem in Two and Three Dimensions
type: item
description: In this paper, we analyze the virtual element method for the symmetric second order elliptic eigenvalue problem with variable coefficients in two and three dimensions, which reduces the number of degrees of freedom of the standard virtual element method. We attempt to prove the interpolation theory and stability analysis for the serendipity nodal virtual element space, which provides new stabilization terms for the virtual element schemes. Then we prove the spectral approximation and the optimal a priori error estimates. Moreover, we construct a fully computable residual-type a posteriori error estimator applied to the adaptive serendipity virtual element method and prove its upper and lower bounds with respect to the approximation error. Finally, we show numerical examples to verify the theoretical results and show the comparison between standard and serendipity virtual element methods.
subject: Articles
  topic: section
subject: Serendipity virtual element methods
  topic: keywords
subject: Self-adjoint elliptic eigenvalue problems
  topic: keywords
subject: Interpolation estimates
  topic: keywords
subject: Stability analysis
  topic: keywords
subject: A priori and a posteriori error estimates
  topic: keywords
creator: Haokun Li
creator: Yulin Liu
creator: Jian Meng
creator: Xu Qian
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-01-20
publication date: 2026-01-20
doi: 10.4208/jcm.2508-m2024-0265

Multi-Level Mixed Finite Element Methods Based on Different Iterations for the Steady Boussinesq Problem
type: item
description: Three multi-level mixed finite element methods for the steady Boussinesq equations are analyzed and discussed in this paper. The nonlinear and multi-variables coupled problem on a coarse mesh with the mesh size $h_0$ is solved firstly, and then, a series of decoupled and linear subproblems with the Stokes, Oseen and Newton iterations are solved on the successive and refined grids with the mesh sizes $h_j, j = 1, 2, . . . , J$. The computational scales are reduced and the computational costs are saved. Furthermore, the uniform stability and convergence results in both $L^2$- and $H^1$-norms of are derived under some uniqueness conditions by using the mathematical induction and constructing the dual problems. Theoretical results show that the multi-level methods have the same order of numerical solutions in the $H^1$-norm as the one level method with the mesh sizes $h_j = h^2_{j−1}$, $j = 1, 2, . . . , J$. Finally, some numerical results are provided to investigate and compare the effectiveness of the multi-level mixed finite element methods.
subject: Articles
  topic: section
subject: Steady Boussinesq problem
  topic: keywords
subject: Multi-level methods
  topic: keywords
subject: Stability
  topic: keywords
subject: Error estimates
  topic: keywords
creator: Liwei Liu
creator: Tong Zhang
creator: Chuanjun Chen
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-01-13
publication date: 2026-01-13
doi: 10.4208/jcm.2508-m2024-0257

A Low Order Globally Divergence-Free WG Finite Element Method for Steady Thermally Coupled Incompressible MHD Flow
type: item
description: This paper develops a low order weak Galerkin (WG) finite element method for the steady thermally coupled incompressible magnetohydrodynamics flow. In the interior of elements, the WG scheme uses piecewise linear polynomials for the approximations of the velocity, the magnetic field and the temperature, and piecewise constants for the approximations of the pressure and the magnetic pseudo-pressure; and on the interfaces of elements, the scheme uses piecewise constants for the numerical traces of velocity and the temperature, and piecewise linear polynomials for the numerical traces of the magnetic fields, the pressure and the magnetic pseudo-pressure. This WG method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. Existence and uniqueness results as well as optimal a priori error estimates for the discrete scheme are obtained. A convergent linearized iterative algorithm is presented. Numerical experiments are provided to verify the theoretical analysis.
subject: Articles
  topic: section
subject: Thermally coupled incompressible magnetohydrodynamics flow
  topic: keywords
subject: Weak Galerkin method
  topic: keywords
subject: Globally divergence-free
  topic: keywords
subject: Error estimate
  topic: keywords
creator: Min Zhang
creator: Qijia Zhai
creator: Xiaoping Xie
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-01-10
publication date: 2026-01-10
doi: 10.4208/jcm.2506-m2025-0061

A Novel TGMFECN Dimension Reduction Method Based on POD for the Improved Nonlinear Fourth-Order Cahn-Hilliard Equations
type: item
description: An improved nonlinear fourth-order Cahn-Hilliard (INFOCH) equation is first developed to ensure that its numerical model is symmetric, positive definite, and solvable. Then, by introducing an auxiliary function, the INFOCH equation is decomposed into the nonlinear system of equations with second-order derivatives of spatial variables. Subsequently, by using the Crank-Nicolson (CN) technique to discretize the time derivative, a new time semi-discretized mixed CN (TSDMCN) scheme with second-order accuracy is constructed, and the existence, stability, and error estimates of TSDMCN solutions are analyzed. Thenceforth, a new two-grid mixed finite element (MFE) CN (TGMFECN) method is created by using two-grid MFE method to discretize the TSDMCN scheme, and the existence, stability, and error estimates of TGMFECN solutions are discussed. Next, it is most important that by using proper orthogonal decomposition to reduce the dimension of unknown coefficient vectors of TGMFECN solutions and keep the MFE basis functions unchanged, a new TGMFECN dimensionality reduction (TGMFECNDR) method with very few unknowns, unconditional stability, and second-order time precision is created, and the existence, stability, and error estimates of TGMFECNDR solutions are proved. Finally, the superiority of TGMFECNDR method and the correctness of the obtained theoretical results are showed by two sets of numerical experiments.
subject: Articles
  topic: section
subject: The improved nonlinear fourth-order Cahn-Hilliard equation
  topic: keywords
subject: Proper orthogonal decomposition
  topic: keywords
subject: Time semi-discrete mixed Crank-Nicolson scheme
  topic: keywords
subject: Two-grid mixed finite element Crank-Nicolson method
  topic: keywords
subject: Two-grid mixed finite element Crank-Nicolson dimension reduction method
  topic: keywords
subject: Existence
  topic: keywords
subject: stability
  topic: keywords
subject: and error estimate
  topic: keywords
creator: Huanrong Li
creator: Yuejie Li
creator: Liang He
creator: Zhendong Luo
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2025-12-28
publication date: 2025-12-28
doi: 10.4208/jcm.2506-m2024-0245

Low-Degree $\mathcal{O}(h^2)$-Accurate Conservative Finite Element Schemes for Incompressible Stokes Problems on General Triangulations
type: item
description: In this paper, two $\mathcal{O}(h^2)$-accurate conservative finite element schemes with low-degree polynomials for the incompressible Stokes equations are presented. The schemes use respective $H({\rm div})$ finite element spaces, namely the third-order Brezzi-Douglas-Marini space and Brezzi-Douglas-Fortin-Marini space, with enhanced smoothness for the velocity and piecewise quadratic polynomials for the pressure, and are denoted as ${\rm sBDM}_3−{\rm P}_2$ and ${\rm sBDFM}_3−{\rm P}_2$ schemes, respectively. The discrete Korn inequality holds for both ${\rm sBDM}_3$ and ${\rm sBDFM}_3$ finite element spaces. For the ${\rm sBDM}_3−{\rm P}_2$ scheme, the inf-sup condition holds on general triangulations, and for the ${\rm sBDFM}_3−{\rm P}_2$ scheme, the inf-sup condition holds on triangulations with mild restriction. Both schemes achieve an energy norm of velocity errors of $\mathcal{O}(h^2)$ order and an $L^2$-norm of pressure errors of $\mathcal{O}(h^2)$ order. Numerical experiments support the theoretical constructions.
subject: Articles
  topic: section
subject: Stokes equations
  topic: keywords
subject: Strictly conservative finite element
  topic: keywords
subject: Low-degree
  topic: keywords
subject: $\mathcal{O}$($h^2$)-accurate
  topic: keywords
subject: inf-sup stability
  topic: keywords
creator: Wenjia Liu
creator: Huilan Zeng
creator: Shuo Zhang
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2025-12-28
publication date: 2025-12-28
doi: 10.4208/jcm.2509-m2024-0165

Mean-Square Convergence of Two-Step Milstein Methods for Nonlinear Stochastic Delay Differential Equations
type: item
description: In this paper, a numerical method for solving nonlinear stochastic delay differential equations is proposed: two-step Milstein method. The mean square consistent and mean-square convergence of the numerical method are studied. Through the relevant derivation, the conditions that the coefficients need to be satisfied when the numerical method is mean-square consistent and mean-square convergent are obtained, and it is proved that the mean-square convergence order of the numerical method is 1. Finally, the theoretical results are verified by numerical experiments.
subject: Articles
  topic: section
subject: Stochastic delay differential equation
  topic: keywords
subject: Two-step Milstein method
  topic: keywords
subject: Mean-square consistent
  topic: keywords
subject: Mean-square convergence
  topic: keywords
creator: Lijuan Peng
creator: Lihang Zhou
creator: Wenqiang Wang
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2026-04-15
publication date: 2026-04-15
starting page: 578
ending page: 592
doi: 10.4208/jcm.2509-m2025-0023

On the Adaptive Deterministic Block Coordinate Descent Method with Momentum for Solving Large Linear Least-Squares Problems
type: item
description: Inspired by Polyak’s heavy-ball method, this paper proposes an adaptive deterministic block coordinate descent method with momentum (mADBCD) for efficiently solving large-scale linear least-squares problems. The proposed method introduces a novel column selection criterion based on the Euclidean norm of the residual vector of the normal equation. In contrast to classical block coordinate descent methods, mADBCD does not require a fixed pre-partitioning of the column indices of the coefficient matrix and avoids the expensive computation of Moore Penrose pseudoinverses of submatrices at each iteration. The method adaptively updates the block index set at each step, thereby improving both flexibility and scalability. When the coefficient matrix is of full column rank, we prove that mADBCD converges linearly to the unique solution of the least-squares problem. Numerical experiments are conducted to show that mADBCD outperforms several recent block coordinate descent methods in terms of iteration count and CPU time. In particular, when solving extremely sparse least-squares problems, mADBCD is the first block coordinate descent method reported to achieve CPU time nearly comparable to that of the classical least squares QR (LSQR) method [Paige and Saunders, ACM Trans. Math. Softw., 8 (1982)].
subject: Articles
  topic: section
subject: Linear least-squares problem
  topic: keywords
subject: Coordinate descent method
  topic: keywords
subject: Block coordinate descent method
  topic: keywords
subject: Heavy ball method, Convergence
  topic: keywords
creator: Longze Tan
creator: Mingyu Deng
creator: Jiali Qiu
creator: Xueping Guo
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2025-12-10
publication date: 2025-12-10
doi: 10.4208/jcm.2506-m2025-0047

Stochastic Variance Reduced Gradient for Tensor Recovery
type: item
description: Low-rank tensor recovery is pivotal in numerous applications, including image and video processing, machine learning, and data analysis. A common approach to this problem involves convex relaxation, where the tensor rank function is minimized by using the tensor nuclear norm. However, this method can be significantly suboptimal. In addition, the stochastic variance reduced gradient (SVRG) method, a variant of stochastic gradient descent, has been applied to matrix recovery problems. In this paper, we extend the SVRG method to the tensor framework, introducing the tensor stochastic variance reduced gradient (TSVRG) algorithm for tensor recovery with CP or Tucker rank constraints. TSVRG is designed to achieve higher precision solutions by escaping local minima and identifying superior global optima. Moreover, TSVRG offers reduced computational complexity compared to traditional gradient descent methods. We establish a convergence theorem for TSVRG under the tensor restricted isometry condition when the measurements are linear. Finally, we present numerical results using both synthetic and real data, demonstrating the competitive performance of TSVRG compared to other advanced algorithms.
subject: Articles
  topic: section
subject: Tensor recovery
  topic: keywords
subject: Stochastic variance reduced gradient
  topic: keywords
subject: CP rank
  topic: keywords
subject: Tucker rank
  topic: keywords
creator: Li Li
creator: Chen Xu
creator: Jian Lu
creator: Ningning Han
creator: Lixin Shen
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2025-12-05
publication date: 2025-12-05
doi: 10.4208/jcm.2508-m2024-0288

Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation
type: item
description: In this paper, we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation, and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution. We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time $t = 0$. More precisely, we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to $C^2([0,T])$ in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness, otherwise the solution exhibits the same singular behavior like its constant-order counterpart. Based on these regularity results, we prove optimal-order convergence rate of the Galerkin finite element scheme. Furthermore, we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives. Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.
subject: Articles
  topic: section
subject: Variable-order
  topic: keywords
subject: Multi-term time-fractional diffusion equation
  topic: keywords
subject: Solution regularity
  topic: keywords
subject: Galerkin finite element
  topic: keywords
subject: Parareal method.
  topic: keywords
subject: 35B65
  topic: subjects
subject: 35R11
  topic: subjects
subject: 65M12
  topic: subjects
subject: 65M60
  topic: subjects
creator: Huan Liu
creator: Xiangcheng Zheng
creator: Hongfei Fu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2022-10-06
publication date: 2022-10-06
starting page: 814
ending page: 834
doi: 10.4208/jcm.2102-m2020-0211

A Characteristic Mixed Finite Element Two-Grid Method for Compressible Miscible Displacement Problem
type: item
description: A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium. The concentration equation is treated by a mixed finite element method with characteristics (CMFEM) and the pressure equation is treated by a parabolic mixed finite element method (PMFEM). Two-grid algorithm is considered to linearize nonlinear coupled system of two parabolic partial differential equations. Moreover, the $L^q$ error estimates are conducted for the pressure, Darcy velocity and concentration variables in the two-grid solutions. Both theoretical analysis and numerical experiments are presented to show that the two-grid algorithm is very effective.
subject: Articles
  topic: section
subject: Two-grid method
  topic: keywords
subject: Miscible displacement problem
  topic: keywords
subject: Mixed finite element
  topic: keywords
subject: Characteristic finite element method.
  topic: keywords
subject: 65M06
  topic: subjects
subject: 65M12
  topic: subjects
subject: 65M15
  topic: subjects
subject: 65M55
  topic: subjects
creator: Hanzhang Hu
creator: Yanping Chen
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2022-10-06
publication date: 2022-10-06
starting page: 794
ending page: 813
doi: 10.4208/jcm.2101-m2020-0277

Primal-Dual Path-Following Methods and the Trust-Region Updating Strategy for Linear Programming with Noisy Data
type: item
description: In this article, we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem. For the rank-deficient problem with the small noisy data, we also give the preprocessing method based on the QR decomposition with column pivoting. Then, we prove the global convergence of the new method when the initial point is strictly primal-dual feasible. Finally, for some rank-deficient problems with or without the small noisy data from the NETLIB collection, we compare it with other two popular interior-point methods, i.e. the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment. Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data.
subject: Articles
  topic: section
subject: Continuation Newton method
  topic: keywords
subject: Trust-region method
  topic: keywords
subject: Linear programming
  topic: keywords
subject: Rank deficiency
  topic: keywords
subject: Path-following method
  topic: keywords
subject: Noisy data.
  topic: keywords
subject: 65L20
  topic: subjects
subject: 65K05
  topic: subjects
subject: 65L05
  topic: subjects
creator: Xinlong Luo
creator: Yiyan Yao
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2022-10-06
publication date: 2022-10-06
starting page: 756
ending page: 776
doi: 10.4208/jcm.2101-m2020-0173

Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data
type: item
description: This article concerns numerical approximation of a parabolic interface problem with general $L^2$ initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the $k$-step backward difference formula with $ k=1,\ldots,6 $. To maintain high-order convergence in time for possibly nonsmooth $L^2$ initial value, we modify the standard backward difference formula at the first $k-1$ time levels by using a method recently developed for fractional evolution equations. An error bound of $\mathcal{O}(t_n^{-k}\tau^k+t_n^{-1}h^2|\log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data.
subject: Articles
  topic: section
subject: Parabolic interface problem
  topic: keywords
subject: Finite element method
  topic: keywords
subject: Backward difference formulae
  topic: keywords
subject: Error estimate
  topic: keywords
subject: Nonsmooth initial data.
  topic: keywords
subject: 65M60
  topic: subjects
subject: 65N30
  topic: subjects
subject: 65N15
  topic: subjects
creator: Kai Wang
creator: Na Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2022-10-06
publication date: 2022-10-06
starting page: 777
ending page: 793
doi: 10.4208/jcm.2101-m2020-0075

Heavy Ball Flexible GMRES Method for Nonsymmetric Linear Systems
type: item
description: Flexible GMRES (FGMRES) is a variant of preconditioned GMRES, which changes preconditioners at every Arnoldi step. GMRES often has to be restarted in order to save storage and reduce orthogonalization cost in the Arnoldi process. Like restarted GMRES, FGMRES may also have to be restarted for the same reason. A major disadvantage of restarting is the loss of convergence speed. In this paper, we present a heavy ball flexible GMRES method, aiming to recoup some of the loss in convergence speed in the restarted flexible GMRES while keep the benefit of limiting memory usage and controlling orthogonalization cost. Numerical tests often demonstrate superior performance of the proposed heavy ball FGMRES to the restarted FGMRES.
subject: Articles
  topic: section
subject: GMRES
  topic: keywords
subject: Flexible GMRES
  topic: keywords
subject: Heavy ball GMRES
  topic: keywords
subject: Preconditioner
  topic: keywords
subject: Linear system.
  topic: keywords
subject: 65F10
  topic: subjects
creator: Mei Yang
creator: Ren-Cang Li
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2022-10-06
publication date: 2022-10-06
starting page: 711
ending page: 727
doi: 10.4208/jcm.2101-m2019-0243

Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations
type: item
description: Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous $P_1$ vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and $L^2$-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of $L^2$-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis.
subject: Articles
  topic: section
subject: Stationary Navier-Stokes equations
  topic: keywords
subject: Nonconforming finite elements
  topic: keywords
subject: Penalty stabilization methods
  topic: keywords
subject: DG methods
  topic: keywords
subject: Locally divergence-free
  topic: keywords
subject: 65N30
  topic: subjects
subject: 76D05
  topic: subjects
creator: Linshuang He
creator: Minfu Feng
creator: Qiang Ma
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2022-10-06
publication date: 2022-10-06
starting page: 728
ending page: 755
doi: 10.4208/jcm.2101-m2020-0156

A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations
type: item
description: To reduce the computational cost, we propose a regularizing modified Levenberg-Marquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.
subject: Articles
  topic: section
subject: The regularizing Levenberg-Marquardt scheme
  topic: keywords
subject: Multiscale Galerkin methods
  topic: keywords
subject: Nonlinear ill-posed problems
  topic: keywords
subject: Heuristic parameter choice rule
  topic: keywords
subject: Optimal convergence rate.
  topic: keywords
subject: 65J15
  topic: subjects
subject: 65J20
  topic: subjects
subject: 65J22
  topic: subjects
creator: Rong Zhang
creator: Hongqi Yang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2022-10-06
publication date: 2022-10-06
starting page: 686
ending page: 710
doi: 10.4208/jcm.2101-m2020-0218

Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations
type: item
description: In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by $H^{1}$-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice $h=H^2$.
subject: Articles
  topic: section
subject: Semilinear parabolic integro-differential equations
  topic: keywords
subject: H$^1$-Galerkin mixed finite element method
  topic: keywords
subject: A priori error estimates
  topic: keywords
subject: Two-grid
  topic: keywords
subject: Superclose.
  topic: keywords
subject: 49J20
  topic: subjects
subject: 65N30
  topic: subjects
creator: Tianliang Hou
creator: Chunmei Liu
creator: Chunlei Dai
creator: Luoping Chen
creator: Yin Yang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2022-10-06
publication date: 2022-10-06
starting page: 667
ending page: 685
doi: 10.4208/jcm.2101-m2019-0159

A Simple Error Estimate of Discontinuous Galerkin Methods for Elliptic Equations with Low Regularity
type: item
description: In this work, we develop a low-regularity error analysis for the interior-penalty discontinuous Galerkin (IPDG) method, incorporating numerical fluxes originally proposed by Brezzi et al. [Numer. Methods Partial Differential Equations, 16 (2000)]. Our analysis specifically addresses elliptic problems with solutions residing in the low-regularity space $H^s,$ where $0 ≤ s < 1/2.$ Notably, our error estimates hold under two critical settings: discontinuous coefficients and general Lipschitz domains, precisely capturing the essential features of practical applications. We establish error estimates in the energy norm and the $L^2$-norm, providing a complete theoretical framework for the IPDG method in low-regularity limitation. To systematically verify the theoretical results, we conduct some numerical experiments incorporating precision-controlled parameters that directly correspond to the analytical model’s constraints.
subject: Articles
  topic: section
subject: Discontinuous Galerkin
  topic: keywords
subject: Second-order elliptic equations
  topic: keywords
subject: Minimal regularity
  topic: keywords
subject: Lifting operator
  topic: keywords
creator: Kaifang Liu
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2025-11-26
publication date: 2025-11-26
doi: 10.4208/jcm.2506-m2025-0066

Inverse Nodal Problem for the Sturm-Liouville Operator with Three Discontinuities
type: item
description: In this work, the inverse nodal problem for the Sturm-Liouville operator with three discontinuities is studied. It is proved that the dense nodes of the eigenfunctions can uniquely determine the potential on the whole interval and some parameters, and a reconstruction algorithm for the solution is presented. Finally, numerical examples were provided, and the effectiveness of the algorithm was verified through numerical calculations.
subject: Articles
  topic: section
subject: Inverse nodal problems
  topic: keywords
subject: Discontinuities
  topic: keywords
subject: Uniqueness theorem
  topic: keywords
subject: Reconstruction algorithm
  topic: keywords
creator: Ran Zhang
creator: Xin-Jian Xu
creator: Chuan-Fu Yang
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2025-11-26
publication date: 2025-11-26
doi: 10.4208/jcm.2510-m2025-0196

A Second-Order in Time and Energy-Dissipative Scheme for Time-Fractional Navier-Stokes Equations
type: item
description: This paper proposes an energy-dissipative scheme for solving two- and three-dimensional time-fractional Navier-Stokes equations. The numerical scheme is constructed, using nonuniform $L2−1σ$ approximation in the temporal direction and the Fourier spectral method in the spatial direction. It is shown that the numerical scheme can keep discrete energy stable and the numerical solutions are uniformly bounded without any restriction on step sizes. Error estimates of the fully-discrete scheme are presented. Moreover, a fast algorithm is applied to accelerate the computation. Numerical results in long time intervals are presented to confirm the effectiveness and high efficiency of the scheme.
subject: Articles
  topic: section
subject: Nonuniform $L2−1σ$ scheme
  topic: keywords
subject: Time-fractional Navier-Stokes equation
  topic: keywords
subject: Energy dissipation
  topic: keywords
subject: Error estimates
  topic: keywords
subject: Adaptive time stepping
  topic: keywords
creator: Ruimin Gao
creator: Dongfang Li
creator: Hongyu Qin
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2025-11-26
publication date: 2025-11-26
doi: 10.4208/jcm.2505-m2024-0212

Compact Finite Difference Scheme for Some Sobolev Type Equations with Dirichlet Boundary Condition
type: item
description: This study aims to construct a stable, high-order compact finite difference method for solving Sobolev-type equations with Dirichlet boundary conditions. Approximation of higher-order mixed derivatives in some specific Sobolev-type equations requires a bigger stencil information. One can approximate such derivatives on compact stencils, which are higher-order accurate and take less stencil information but are implicit and sparse. Spatial derivatives in this work are approximated using the sixth-order compact finite difference method, while temporal derivatives are handled with the explicit forward Euler difference scheme. We examine the accuracy and convergence behavior of the proposed scheme. Using the von Neumann stability analysis, we establish $L_2$-stability theory for the linear case. We derive conditions under which fully discrete schemes are stable. Also, the amplification factor C($θ$) is analyzed to ensure the decay property over time. Real parts of C($θ$) lying on the negative real axis confirm the exponential decay of the solution. A series of numerical experiments were performed to verify the effectiveness of the proposed scheme. These tests include both one-dimensional and two-dimensional cases of advection-free and advection-diffusion flows. They also cover applications to the equal width equation, such as the propagation of a single solitary wave, interactions between two and three solitary waves, undular bore formation, and the Benjamin-Bona-Mahony-Burgers equation.
subject: Articles
  topic: section
subject: High-order compact scheme
  topic: keywords
subject: Wave propagation
  topic: keywords
subject: Stability
  topic: keywords
subject: Equal-width equation
  topic: keywords
subject: Benjamin-Bona-Mahony-Burgers equation
  topic: keywords
creator: Lavanya V Salian
creator: Rathan Samala
creator: Rakesh Kumar
rights: Copyright (c) 2025 Journal of Computational Mathematics
date: 2025-11-19
publication date: 2025-11-19
doi: 10.4208/jcm.2509-m2024-0296