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    ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR FULLY DISCRETE SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS
    Ram Manohar, Rajen Kumar Sinha
    Journal of Computational Mathematics    2022, 40 (2): 147-176.   DOI: 10.4208/jcm.2009-m2019-0194
    Abstract74)      PDF
    This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the L (0, T; L 2(Ω))-norm. Finally, a numerical experiment is performed to illustrate the performance of the derived estimators.
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    NUMERICAL ANALYSIS OF A NONLINEAR SINGULARLY PERTURBED DELAY VOLTERRA INTEGRO-DIFFERENTIAL EQUATION ON AN ADAPTIVE GRID
    Libin Liu, Yanping Chen, Ying Liang
    Journal of Computational Mathematics    2022, 40 (2): 258-274.   DOI: 10.4208/jcm.2008-m2020-0063
    Abstract61)      PDF
    In this paper, we study a nonlinear first-order singularly perturbed Volterra integrodifferential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.
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    STOCHASTIC TRUST-REGION METHODS WITH TRUST-REGION RADIUS DEPENDING ON PROBABILISTIC MODELS
    Xiaoyu Wang, Ya-xiang Yuan
    Journal of Computational Mathematics    2022, 40 (2): 294-334.   DOI: 10.4208/jcm.2012-m2020-0144
    Abstract53)      PDF
    We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic models. Especially, we propose a specific algorithm termed STRME, in which the trust-region radius depends linearly on the gradient used to define the latest model. The complexity results of the STRME method in nonconvex, convex and strongly convex settings are presented, which match those of the existing algorithms based on probabilistic properties. In addition, several numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust-region methods and other relevant stochastic gradient methods.
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    CONVERGENCE AND MEAN-SQUARE STABILITY OF EXPONENTIAL EULER METHOD FOR SEMI-LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS
    Haiyan Yuan
    Journal of Computational Mathematics    2022, 40 (2): 177-204.   DOI: 10.4208/jcm.2010-m2019-0200
    Abstract46)      PDF
    In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order 1/2 and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.
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    ON DISTRIBUTED H 1 SHAPE GRADIENT FLOWS IN OPTIMAL SHAPE DESIGN OF STOKES FLOWS: CONVERGENCE ANALYSIS AND NUMERICAL APPLICATIONS
    Jiajie Li, Shengfeng Zhu
    Journal of Computational Mathematics    2022, 40 (2): 231-257.   DOI: 10.4208/jcm.2009-m2020-0020
    Abstract35)      PDF
    We consider optimal shape design in Stokes flow using H 1 shape gradient flows based on the distributed Eulerian derivatives. MINI element is used for discretizations of Stokes equation and Galerkin finite element is used for discretizations of distributed and boundary H 1 shape gradient flows. Convergence analysis with a priori error estimates is provided under general and different regularity assumptions. We investigate the performances of shape gradient descent algorithms for energy dissipation minimization and obstacle flow. Numerical comparisons in 2D and 3D show that the distributed H 1 shape gradient flow is more accurate than the popular boundary type. The corresponding distributed shape gradient algorithm is more effective.
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    A θ-L APPROACH FOR SOLVING SOLID-STATE DEWETTING PROBLEMS
    Weijie Huang, Wei Jiang, Yan Wang
    Journal of Computational Mathematics    2022, 40 (2): 275-293.   DOI: 10.4208/jcm.2010-m2020-0040
    Abstract33)      PDF
    We propose a θ-L approach for solving a sharp-interface model about simulating solidstate dewetting of thin films with isotropic/weakly anisotropic surface energies. The sharpinterface model is governed by surface diffusion and contact line migration. For solving the model, traditional numerical methods usually suffer from the severe stability constraint and/or the mesh distribution trouble. In the θ-L approach, we introduce a useful tangential velocity along the evolving interface and utilize a new set of variables (i.e., the tangential angle θ and the total length L of the interface curve), so that it not only could reduce the stiffness resulted from the surface tension, but also could ensure the mesh equidistribution property during the evolution. Furthermore, it can achieve second-order accuracy when implemented by a semi-implicit linear finite element method. Numerical results are reported to demonstrate that the proposed θ-L approach is efficient and accurate.
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    UNCONDITIONALLY OPTIMAL ERROR ESTIMATES OF THE BILINEAR-CONSTANT SCHEME FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS
    Huaijun Yang, Dongyang Shi
    Journal of Computational Mathematics    2022, 40 (1): 127-146.   DOI: 10.4208/jcm.2007-m2020-0164
    Abstract28)      PDF
    In this paper, the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme. The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally, while the previous works require certain time-step restrictions. The analysis is based on an iterated timediscrete system, with which the error function is split into a temporal error and a spatial error. The τ-independent ( τ is the time stepsize) error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis, which implies that the numerical solution in L -norm is bounded. Thus optimal error estimates can be obtained in a traditional way. Numerical results are provided to confirm the theoretical analysis.
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    A FINITE ELEMENT ALGORITHM FOR NEMATIC LIQUID CRYSTAL FLOW BASED ON THE GAUGE-UZAWA METHOD
    Pengzhan Huang, Yinnian He, Ting Li
    Journal of Computational Mathematics    2022, 40 (1): 26-43.   DOI: 10.4208/jcm.2005-m2020-0010
    Abstract28)      PDF
    In this paper, we present a finite element algorithm for the time-dependent nematic liquid crystal flow based on the Gauge-Uzawa method. This algorithm combines the Gauge and Uzawa methods within a finite element variational formulation, which is a fully discrete projection type algorithm, whereas many projection methods have been studied without space discretization. Besides, error estimates for velocity and molecular orientation of the nematic liquid crystal flow are shown. Finally, numerical results are given to show that the presented algorithm is reliable and confirm the theoretical analysis.
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    ADAPTIVE AND OPTIMAL POINT-WISE ESTIMATIONS FOR DENSITIES IN GARCH-TYPE MODEL BY WAVELETS
    Cong Wu, Jinru Wang, Xiaochen Zeng
    Journal of Computational Mathematics    2022, 40 (1): 108-126.   DOI: 10.4208/jcm.2007-m2020-0109
    Abstract27)      PDF
    This paper considers adaptive point-wise estimations of density functions in GARCHtype model under the local Hölder condition by wavelet methods. A point-wise lower bound estimation of that model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate, which means that the convergence rate coincides with the lower bound. The non-linear wavelet estimator is introduced for adaptivity, although it is nearly-optimal. However, the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data driven version without any assumptions on the estimated functions.
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    CONSTRUCTION OF CUBATURE FORMULAS VIA BIVARIATE QUADRATIC SPLINE SPACES OVER NON-UNIFORM TYPE-2 TRIANGULATION
    Jiang Qian, Xiquan Shi, Jinming Wu, Dianxuan Gong
    Journal of Computational Mathematics    2022, 40 (2): 205-230.   DOI: 10.4208/jcm.2008-m2020-0077
    Abstract24)      PDF
    In this paper, matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in S 2 1mn (2)), and coefficients of splines in terms of B-net are calculated firstly. Moreover, by means of coefficients in terms of B-net, computation of bivariate numerical cubature over triangular sub-domains with respect to variables x and y is transferred into summation of coefficients of splines in terms of B-net. Thus concise bivariate cubature formulas are constructed over rectangular sub-domain. Furthermore, by means of module of continuity and max-norms, error estimates for cubature formulas are derived over both sub-domains and the domain.
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    TWO-GRID ALGORITHM OF H 1-GALERKIN MIXED FINITE ELEMENT METHODS FOR SEMILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS
    Tianliang Hou, Chunmei Liu, Chunlei Dai, Luoping Chen, Yin Yang
    Journal of Computational Mathematics    2022, 40 (5): 667-685.   DOI: 10.4208/jcm.2101-m2019-0159
    Abstract23)      PDF
    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.
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    Cited: CSCD(1)
    DATA-DRIVEN TIGHT FRAME CONSTRUCTION FOR IMPULSIVE NOISE REMOVAL
    Yang Chen, Chunlin Wu
    Journal of Computational Mathematics    2022, 40 (1): 89-107.   DOI: 10.4208/jcm.2008-m2018-0092
    Abstract22)      PDF
    The method of data-driven tight frame has been shown very useful in image restoration problems. We consider in this paper extending this important technique, by incorporating L 1 data fidelity into the original data-driven model, for removing impulsive noise which is a very common and basic type of noise in image data. The model contains three variables and can be solved through an efficient iterative alternating minimization algorithm in patch implementation, where the tight frame is dynamically updated. It constructs a tight frame system from the input corrupted image adaptively, and then removes impulsive noise by the derived system. We also show that the sequence generated by our algorithm converges globally to a stationary point of the optimization model. Numerical experiments and comparisons demonstrate that our approach performs well for various kinds of images. This benefits from its data-driven nature and the learned tight frames from input images capture richer image structures adaptively.
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    LONG-TIME OSCILLATORY ENERGY CONSERVATION OF TOTAL ENERGY-PRESERVING METHODS FOR HIGHLY OSCILLATORY HAMILTONIAN SYSTEMS
    Bin Wang, Xinyuan Wu
    Journal of Computational Mathematics    2022, 40 (1): 70-88.   DOI: 10.4208/jcm.2008-m2018-0218
    Abstract20)      PDF
    For an integrator when applied to a highly oscillatory system, the near conservation of the oscillatory energy over long times is an important aspect. In this paper, we study the long-time near conservation of oscillatory energy for the adapted average vector field (AAVF) method when applied to highly oscillatory Hamiltonian systems. This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly. This paper is devoted to analysing another important property of AAVF method, i.e., the near conservation of its oscillatory energy in a long term. The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion. A similar result of the method in the multi-frequency case is also presented in this paper.
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    METRICALLY REGULAR MAPPING AND ITS UTILIZATION TO CONVERGENCE ANALYSIS OF A RESTRICTED INEXACT NEWTON-TYPE METHOD
    Mohammed Harunor Rashid
    Journal of Computational Mathematics    2022, 40 (1): 44-69.   DOI: 10.4208/jcm.2005-m2019-0019
    Abstract19)      PDF
    In the present paper, we study the restricted inexact Newton-type method for solving the generalized equation 0 ∈ f( x)+ F ( x), where X and Y are Banach spaces, f: XY is a Fréchet differentiable function and F: XY is a set-valued mapping with closed graph. We establish the convergence criteria of the restricted inexact Newton-type method, which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Fréchet derivative of f. Indeed, we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Fréchet derivative of f is continuous and Lipschitz continuous as well as f + F is metrically regular. An application of this method to variational inequality is given. In addition, a numerical experiment is given which illustrates the theoretical result.
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    THEORETICAL ANALYSES ON DISCRETE FORMULAE OF DIRECTIONAL DIFFERENTIALS IN THE FINITE POINT METHOD
    Guixia Lv, Longjun Shen
    Journal of Computational Mathematics    2022, 40 (1): 1-25.   DOI: 10.4208/jcm.2005-m2019-0304
    Abstract19)      PDF
    For the five-point discrete formulae of directional derivatives in the finite point method, overcoming the challenge resulted from scattered point sets and making full use of the explicit expressions and accuracy of the formulae, this paper obtains a number of theoretical results:(1) a concise expression with definite meaning of the complicated directional difference coefficient matrix is presented, which characterizes the correlation between coefficients and the connection between coefficients and scattered geometric characteristics; (2) various expressions of the discriminant function for the solvability of numerical differentials along with the estimation of its lower bound are given, which are the bases for selecting neighboring points and making analysis; (3) the estimations of combinatorial elements and of each element in the directional difference coefficient matrix are put out, which exclude the existence of singularity. Finally, the theoretical analysis results are verified by numerical calculations.
    The results of this paper have strong regularity, which lay the foundation for further research on the finite point method for solving partial differential equations.
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    ANALYSIS OF A MULTI-TERM VARIABLE-ORDER TIME-FRACTIONAL DIFFUSION EQUATION AND ITS GALERKIN FINITE ELEMENT APPROXIMATION
    Huan Liu, Xiangcheng Zheng, Hongfei Fu
    Journal of Computational Mathematics    2022, 40 (5): 814-834.   DOI: 10.4208/jcm.2102-m2020-0211
    Abstract8)      PDF
    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 optimalorder 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.
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    Cited: CSCD(1)
    PENALTY-FACTOR-FREE STABILIZED NONCONFORMING FINITE ELEMENTS FOR SOLVING STATIONARY NAVIER-STOKES EQUATIONS
    Linshuang He, Minfu Feng, Qiang Ma
    Journal of Computational Mathematics    2022, 40 (5): 728-755.   DOI: 10.4208/jcm.2101-m2020-0156
    Abstract7)      PDF
    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.
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    Cited: CSCD(1)
    STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR A CLASS OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS
    Wei Zhang
    Journal of Computational Mathematics    2022, 40 (4): 607-623.   DOI: 10.4208/jcm.2101-m2020-0070
    Abstract7)      PDF
    In this paper, we consider the Euler-Maruyama method for a class of stochastic Volterra integral equations (SVIEs). It is known that the strong convergence order of the Euler-Maruyama method is $\frac{1}{2}$. However, the strong superconvergence order 1 can be obtained for a class of SVIEs if the kernels σ i( t, t) = 0 for i = 1 and 2; otherwise, the strong convergence order is $\frac{1}{2}$ . Moreover, the theoretical results are illustrated by some numerical examples.
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    Cited: CSCD(1)
    DIRECT IMPLEMENTATION OF TIKHONOV REGULARIZATION FOR THE FIRST KIND INTEGRAL EQUATION
    Meisam Jozi, Saeed Karimi
    Journal of Computational Mathematics    2022, 40 (3): 335-353.   DOI: 10.4208/jcm.2010-m2020-0132
    Abstract7)      PDF
    A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations, is first to discretize and then to work with the final linear system. This unavoidably inflicts discretization errors which may lead to disastrous results, especially when a quadrature rule is used. We propose to regularize directly the integral equation resulting in a continuous Tikhonov problem. The Tikhonov problem is reduced to a simple least squares problem by applying the Golub-Kahan bidiagonalization (GKB) directly to the integral operator. The regularization parameter and the iteration index are determined by the discrepancy principle approach. Moreover, we study the discrete version of the proposed method resulted from numerical evaluating the needed integrals. Focusing on the nodal values of the solution results in a weighted version of GKB-Tikhonov method for linear systems arisen from the Nyström discretization. Finally, we use numerical experiments on a few test problems to illustrate the performance of our algorithms.
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    Cited: CSCD(1)
    BOUNDARY INTEGRAL EQUATIONS FOR ISOTROPIC LINEAR ELASTICITY
    Benjamin Stamm, Shuyang Xiang
    Journal of Computational Mathematics    2022, 40 (6): 835-864.   DOI: 10.4208/jcm.2103-m2019-0031
    Abstract7)      PDF
    This articles first investigates boundary integral operators for the three-dimensional isotropic linear elasticity of a biphasic model with piecewise constant Lamé coefficients in the form of a bounded domain of arbitrary shape surrounded by a background material. In the simple case of a spherical inclusion, the vector spherical harmonics consist of eigenfunctions of the single and double layer boundary operators and we provide their spectra. Further, in the case of many spherical inclusions with isotropic materials, each with its own set of Lamé parameters, we propose an integral equation and a subsequent Galerkin discretization using the vector spherical harmonics and apply the discretization to several numerical test cases.
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