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    15 July 2021, Volume 39 Issue 4
    ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS
    Tamal Pramanick
    2021, 39(4):  493-517.  DOI: 10.4208/jcm.2001-m2019-0117
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    We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane. This new class of finite elements, which is called composite finite elements, was first introduced by Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] for the approximation of partial differential equations on domains with complicated geometry. The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving partial differential equations by domain discretization method. The composite finite element method introduces two-scale grid for discretization of the domain, the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only. While the fine-scale grid is used to resolve the Dirichlet boundary condition, the dimension of the finite element space depends only on the coarse-scale grid. As a consequence, the resulting linear system will have a fewer number of unknowns. A continuous, piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods. We have derived the error estimates in the L(L2)-norm for both semidiscrete and fully discrete schemes. Moreover, numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.
    SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS
    Yong Liu, Chi-Wang Shu, Mengping Zhang
    2021, 39(4):  518-537.  DOI: 10.4208/jcm.2003-m2018-0305
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    In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the L2-norm for even degree polynomial approximations. On uniform meshes, the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions, improving previous results. The theoretical findings are found to be sharp and consistent with numerical results.
    SCHWARZ METHOD FOR FINANCIAL ENGINEERING
    Guangbao Guo, Weidong Zhao
    2021, 39(4):  538-555.  DOI: 10.4208/jcm.2003-m2018-0115
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    Schwarz method is put forward to solve second order backward stochastic differential equations (2BSDEs) in this work. We will analyze uniqueness, convergence, stability and optimality of the proposed method. Moreover, several simulation results are presented to demonstrate the effectiveness; several applications of the 2BSDEs are investigated. It is concluded from these results that the proposed the method is powerful to calculate the 2BSDEs listing from the financial engineering.
    CHARACTERISATION OF RATIONAL AND NURBS DEVELOPABLE SURFACES IN COMPUTER AIDED DESIGN
    Leonardo Fernández-Jambrina
    2021, 39(4):  556-573.  DOI: 10.4208/jcm.2003-m2019-0226
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    In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions Λ, M, ν. Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions Λ, M, ν, which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative. It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant Λ, M, ν . The results are readily extended to rational spline developable surfaces.
    STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE
    Xiaobing Feng, Yukun Li, Yi Zhang
    2021, 39(4):  574-598.  DOI: 10.4208/jcm.2003-m2019-0250
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    This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz condition. These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations (SODEs) under “minimum assumptions” were studied. As a result, the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs. There are several difficulties which need to be overcome for this generalization. First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty. It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix. In this paper we use a finite element interpolation technique to discretize the nonlinear drift term. Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the H1-seminorm of the numerical solution must be established, which are difficult and delicate. A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the goal. Finally, stability estimates for the second and higher order moments of the L2-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant. This is done by utilizing the interpolation theory and the higher moment estimates for the H1-seminorm of the numerical solution. After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence. Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.
    IMPLICIT-EXPLICIT RUNGE-KUTTA-ROSENBROCK METHODS WITH ERROR ANALYSIS FOR NONLINEAR STIFF DIFFERENTIAL EQUATIONS
    Bin Huang, Aiguo Xiao, Gengen Zhang
    2021, 39(4):  599-620.  DOI: 10.4208/jcm.2005-m2019-0238
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    Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear stiff ordinary differential equations by combining linearly implicit Rosenbrock methods with explicit Runge-Kutta methods. First, the general order conditions up to order 3 are obtained. Then, for the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, the corresponding errors of the implicit-explicit methods are analysed. At last, some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the methods.
    CONVERGENCE ANALYSIS ON SS-HOPM FOR BEC-LIKE NONLINEAR EIGENVALUE PROBLEMS
    Yaozong Tang, Qingzhi Yang, Gang Luo
    2021, 39(4):  621-632.  DOI: 10.4208/jcm.2005-m2019-0298
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    Shifted symmetric higher-order power method (SS-HOPM) has been proved effective in solving the nonlinear eigenvalue problem oriented from the Bose-Einstein Condensation (BEC-like NEP for short) both theoretically and numerically. However, the convergence of the sequence generated by SS-HOPM is based on the assumption that the real eigenpairs of BEC-like NEP are finite. In this paper, we will establish the point-wise convergence via Lojasiewicz inequality by introducing a new related sequence.
    ITERATIVE ILU PRECONDITIONERS FOR LINEAR SYSTEMS AND EIGENPROBLEMS
    Daniele Boffi, Zhongjie Lu, Luca F. Pavarino
    2021, 39(4):  633-654.  DOI: 10.4208/jcm.2009-m2020-0138
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    Iterative ILU factorizations are constructed, analyzed and applied as preconditioners to solve both linear systems and eigenproblems. The computational kernels of these novel Iterative ILU factorizations are sparse matrix-matrix multiplications, which are easy and efficient to implement on both serial and parallel computer architectures and can take full advantage of existing matrix-matrix multiplication codes. We also introduce level-based and threshold-based algorithms in order to enhance the accuracy of the proposed Iterative ILU factorizations. The results of several numerical experiments illustrate the efficiency of the proposed preconditioners to solve both linear systems and eigenvalue problems.