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.
A Hybrid Two-Level Finite Element Method for the Stationary Natural Convection Problem and Parallel Implement
This study presents a novel hybrid approach for addressing incompressible stationary natural convection problem, incorporating a parallel technique to enhance computational efficiency. Inspired by the traditional two-level method [He and Wang, Comput. Methods Appl. Mech. Engrg., 197 (2008)] and the two-step approach [Wu et al., Int. J. Heat Mass Transfer, 101 (2016)], both characterized by their iterative and corrective processes, we endeavor to alleviate the computational burden associated with the iterative process. Building upon these methods, our novel hybrid method involves two primary steps: initially solving the original problem using the finite element pair $P_{1}b-P_1-P_1$ on a coarse mesh, followed by resolving the linearized equations using the higher-order pair $P_2-P_1-P_2$ on a fine mesh. While the first step employs iterative techniques, the second step entails directly solving a linearized problem. This novel approach can save lots of computational time in the iterative step compared to the traditional methods. Moreover, leveraging domain decomposition techniques, we implement a parallel strategy to further accelerate computations. Finally, we conduct several numerical examples to validate the efficiency of the proposed algorithms. The numerical results demonstrate optimal convergence rates comparable to those obtained using only the $P_2-P_1-P_2$ finite element pair under similar relative error conditions. Furthermore, the numerical simulations on the two obstacles flow and Bénard convection problem show the robustness and efficiency of the proposed algorithms.
Copyright (c) 2025 Journal of Computational Mathematics
Cubically Convergent Two-Step Gauss-Newton Method for Nonsmooth Equations with Application to AVE
We propose a two-step Gauss-Newton method (TS-GNM) for solving nonsmooth equations. At each iteration, the TS-GNM solves both a Gauss-Newton equation and an approximate Gauss-Newton equation. A second-order derivative-free line search strategy is designed to ensure the global convergence of TS-GNM. Under the nonsingularity condition and the strong semismoothness of the underlying function, we prove that the TS-GNM converges quadratically. Furthermore, we demonstrate that the TS-GNM achieves a cubic convergence rate when the generalized Jacobian is locally Lipschitz continuous at the solutions. Finally, we pay particular attention to the absolute value equation and present some numerical results.
Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation
In this paper, we consider the sparse distributed control problem constrained by a random elliptic equation, which we reformulate as a nonsmooth stochastic optimization problem in Hilbert space. By incorporating the advantages of the stochastic approximation approach and the alternating direction method of multipliers (ADMM), we propose a stochastic ADMM algorithm. This method decouples the stochasticity arising from the random equation constraint from the nonsmoothness of the control objective, allowing them to be tackled separately within the iterations. We introduce stochastic gradients and develop a proximal linearization technique for the stochastic subproblem, allowing each subproblem to admit a closed-form solution. The convergence and a high-probability bound of the proposed method are analyzed for the model problem. Numerical results demonstrate the effectiveness and efficiency of our method.
Alternating direction method of multipliers
The Exponential Scalar Auxiliary Variable Approach for the Landau-Lifshitz Equation
In this paper, we present two structure-preserving numerical schemes for the Landau-Lifshitz equation by combining the exponential scalar auxiliary variable method with the projection method. These schemes preserve both the length constraint and the modified energy dissipation law, ensuring numerical stability and accuracy. Moreover, they are particularly well-suited for studying the Landau-Lifshitz equation with higher-order energy terms, which have often been overlooked in earlier studies but have a significant impact on the stability, dynamics, and thermal behavior of magnetic skyrmions. We establish the unique solvability and energy stability of the schemes, and provide a rigorous error analysis. Numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed schemes.
On the Superconvergence of Collocation Methods Based on Two Post-Processing Techniques for Fractional Integro-Differential Equations with Weakly Singular Kernels
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.
Fractional integro-differential equation
Characterizing the Rate of Convergence of the Augmented Lagrange Method for Nonlinear Programming
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.
An Accelerated Preconditioned Primal-Dual Gradient Algorithm for Nonconvex Composite Optimization Problems with Applications
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.
Finite Element Method with Grünwald-Letnikov Type Approximation in Time for a Constant Time Delay Subdiffusion Equation
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.
Subdiffusion equation with constant time delay
The Quadratic Discontinuous Finite Volume Element Schemes for Elliptic Problems
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.
Discontinuous finite volume element method
An Analysis of the Saturation Assumption for Polynomial Interpolation with Application to Adaptive Finite Element Methods
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.
A Multiple-Invariants Preserving Scheme for Modified Two-Component Euler-Poincaré Equations
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.
Modified Two-Component Euler-Poincaré Equations
High Oder Probabilistic Numerical Methods for Forward Backward Stochastic Differential Equations
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.
Forward backward stochastic differential equation
Regularization Method with Two Differential Operators for Simultaneous Inversion of Source Term and Initial Value in a Time-Fractional Black-Scholes Equation
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.
Regularization method with two differential operators
Simultaneous inversion of source term and initial value
A Structure-Preserving Nonconforming FEM of Nonlinear Kirchhoff-Type Equation with Damping
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.
A structure-preserving nonconforming finite element method
Serendipity Virtual Element Method for the Second Order Elliptic Eigenvalue Problem in Two and Three Dimensions
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.
Self-adjoint elliptic eigenvalue problems
A priori and a posteriori error estimates
Multi-Level Mixed Finite Element Methods Based on Different Iterations for the Steady Boussinesq Problem
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.
A Low Order Globally Divergence-Free WG Finite Element Method for Steady Thermally Coupled Incompressible MHD Flow
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.
Thermally coupled incompressible magnetohydrodynamics flow
A Novel TGMFECN Dimension Reduction Method Based on POD for the Improved Nonlinear Fourth-Order Cahn-Hilliard Equations
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.
The improved nonlinear fourth-order Cahn-Hilliard equation
Time semi-discrete mixed Crank-Nicolson scheme
Two-grid mixed finite element Crank-Nicolson method
Two-grid mixed finite element Crank-Nicolson dimension reduction method
Low-Degree $\mathcal{O}(h^2)$-Accurate Conservative Finite Element Schemes for Incompressible Stokes Problems on General Triangulations
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.
Mean-Square Convergence of Two-Step Milstein Methods for Nonlinear Stochastic Delay Differential Equations
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.
On the Adaptive Deterministic Block Coordinate Descent Method with Momentum for Solving Large Linear Least-Squares Problems
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)].
Stochastic Variance Reduced Gradient for Tensor Recovery
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.
A Characteristic Mixed Finite Element Two-Grid Method for Compressible Miscible Displacement Problem
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.
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Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation
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.
Multi-term time-fractional diffusion equation
Primal-Dual Path-Following Methods and the Trust-Region Updating Strategy for Linear Programming with Noisy Data
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.
Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data
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.
Heavy Ball Flexible GMRES Method for Nonsymmetric Linear Systems
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.
Penalty-Factor-Free Stabilized Nonconforming Finite Elements for Solving Stationary Navier-Stokes Equations
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.
A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations
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.
The regularizing Levenberg-Marquardt scheme
Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations
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 $h=H^2$.
Semilinear parabolic integro-differential equations
H$^1$-Galerkin mixed finite element method