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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
    Abstract84)      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
    Abstract76)      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
    Abstract64)      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
    Abstract55)      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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    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
    Abstract48)      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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    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
    Abstract41)      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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    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
    Abstract33)      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)
    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
    Abstract27)      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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    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
    Abstract16)      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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    Cited: CSCD(1)
    NUMERICAL ANALYSIS OF A PROBLEM INVOLVING A VISCOELASTIC BODY WITH DOUBLE POROSITY
    Noelia Bazarra, José R. Fernández, MariCarme Leseduarte, Antonio Magaña, Ramón Quintanilla
    Journal of Computational Mathematics    2022, 40 (3): 415-436.   DOI: 10.4208/jcm.2010-m2020-0043
    Abstract15)      PDF
    We study from a numerical point of view a multidimensional problem involving a viscoelastic body with two porous structures. The mechanical problem leads to a linear system of three coupled hyperbolic partial differential equations. Its corresponding variational formulation gives rise to three coupled parabolic linear equations. An existence and uniqueness result, and an energy decay property, are recalled. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are performed in one and two dimensions to show the accuracy of the approximation and the behaviour of the solution.
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    Cited: CSCD(1)
    STABILIZED NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY ON RECTANGULAR OR CUBIC MESHES
    Bei Zhang, Jikun Zhao, Minghao Li, Hongru Chen
    Journal of Computational Mathematics    2022, 40 (6): 865-881.   DOI: 10.4208/jcm.2103-m2020-0143
    Abstract14)      PDF
    Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation. The stabilized mixed method is locking-free. The optimal convergence order is derived in the L 2-norm for stress and in the broken H 1-norm and L 2-norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method.
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    Cited: CSCD(1)
    STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR NONLINEAR STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH TIME-DEPENDENT DELAY
    Siyuan Qi, Guangqiang Lan
    Journal of Computational Mathematics    2022, 40 (3): 437-452.   DOI: 10.4208/jcm.2010-m2020-0129
    Abstract13)      PDF
    We consider a nonlinear stochastic Volterra integral equation with time-dependent delay and the corresponding Euler-Maruyama method in this paper. Strong convergence rate (at fixed point) of the corresponding Euler-Maruyama method is obtained when coefficients f and g both satisfy local Lipschitz and linear growth conditions. An example is provided to interpret our conclusions. Our result generalizes and improves the conclusion in[J. Gao, H. Liang, S. Ma, Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay, Appl. Math. Comput., 348 (2019), 385-398.]
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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
    Abstract12)      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)
    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
    Abstract12)      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)
    A DISCRETIZING LEVENBERG-MARQUARDT SCHEME FOR SOLVING NONLIEAR ILL-POSED INTEGRAL EQUATIONS
    Rong Zhang, Hongqi Yang
    Journal of Computational Mathematics    2022, 40 (5): 686-710.   DOI: 10.4208/jcm.2101-m2020-0218
    Abstract12)      PDF
    To reduce the computational cost, we propose a regularizing modified LevenbergMarquardt 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.
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    Cited: CSCD(1)
    PRIMAL-DUAL PATH-FOLLOWING METHODS AND THE TRUST-REGION UPDATING STRATEGY FOR LINEAR PROGRAMMING WITH NOISY DATA
    Xinlong Luo, Yiyan Yao
    Journal of Computational Mathematics    2022, 40 (5): 756-776.   DOI: 10.4208/jcm.2101-m2020-0173
    Abstract11)      PDF
    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 rankdeficient 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.
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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
    Abstract11)      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)
    GENERAL FULL IMPLICIT STRONG TAYLOR APPROXIMATIONS FOR STIFF STOCHASTIC DIFFERENTIAL EQUATIONS*
    Kai Liu, Guiding Gu
    Journal of Computational Mathematics    2022, 40 (4): 541-569.   DOI: 10.4208/jcm.2011-m2019-0174
    Abstract10)      PDF
    In this paper, we present the backward stochastic Taylor expansions for a Ito process, including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions. We construct the general full implicit strong Taylor approximations (including Ito-Taylor and Stratonovich-Taylor schemes) with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations (SSDE) by employing truncations of backward stochastic Taylor expansions. We demonstrate that these schemes will converge strongly with corresponding order 1, 2, 3, . . . . Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2, and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order 2. We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes. The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms. Our numerical experiment show these points.
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    Cited: CSCD(1)
    A DISSIPATION-PRESERVING INTEGRATOR FOR DAMPED OSCILLATORY HAMILTONIAN SYSTEMS
    Wei Shi, Kai Liu
    Journal of Computational Mathematics    2022, 40 (4): 570-588.   DOI: 10.4208/jcm.2011-m2019-0272
    Abstract10)      PDF
    In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of || M||, where M governs the main oscillation of the system and usually || M|| ≫ 1. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.
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    Cited: CSCD(1)
    A NEW HYBRIDIZED MIXED WEAK GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC PROBLEMS*
    Abdelhamid Zaghdani, Sayed Sayari, Miled EL Hajji
    Journal of Computational Mathematics    2022, 40 (4): 499-516.   DOI: 10.4208/jcm.2011-m2019-0142
    Abstract10)      PDF
    In this paper, a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem. This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition. Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established. Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method.
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    Cited: CSCD(1)