Loading...

Table of Content

    15 July 2020, Volume 38 Issue 4
    A TWO-GRID METHOD FOR THE C0 INTERIOR PENALTY DISCRETIZATION OF THE MONGE-AMPERE EQUATION
    Gerard Awanou, Hengguang Li, Eric Malitz
    2020, 38(4):  547-564.  DOI: 10.4208/jcm.1901-m2018-0039
    Asbtract ( 117 )   PDF  
    References | Related Articles | Metrics
    The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a C0 interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal W1,∞ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.
    ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS
    Weifeng Zhang, Shuo Zhang
    2020, 38(4):  565-579.  DOI: 10.4208/jcm.1901-m2018-0150
    Asbtract ( 77 )   PDF  
    References | Related Articles | Metrics
    Quad-curl equations with Navier type boundary conditions are studied in this paper. Stable order reduced formulations equivalent to the model problems are presented, and finite element discretizations are designed. Optimal convergence rates are proved.
    HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY
    Mohammad Tanzil Hasan, Chuanju Xu
    2020, 38(4):  580-605.  DOI: 10.4208/jcm.1902-m2017-0280
    Asbtract ( 49 )   PDF  
    References | Related Articles | Metrics
    In this paper, efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions are investigated.
    THE SHIFTED-INVERSE POWER WEAK GALERKIN METHOD FOR EIGENVALUE PROBLEMS
    Qilong Zhai, Xiaozhe Hu, Ran Zhang
    2020, 38(4):  606-623.  DOI: 10.4208/jcm.1903-m2018-0101
    Asbtract ( 57 )   PDF  
    References | Related Articles | Metrics
    This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique. A high order lower bound can be obtained at a relatively low cost via the proposed method. The error estimates for both eigenvalue and eigenfunction are provided and asymptotic lower bounds are shown as well under some conditions. Numerical examples are presented to validate the theoretical analysis.
    IMPLICITY LINEAR COLLOCATION METHOD AND ITERATED IMPLICITY LINEAR COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF HAMMERSTEIN FREDHOLM INTEGRAL EQUATIONS ON 2D IRREGULAR DOMAINS
    H. Laeli Dastjerdi, M. Nili Ahmadabadi
    2020, 38(4):  624-637.  DOI: 10.4208/jcm.1903-m2017-0206
    Asbtract ( 75 )   PDF  
    References | Related Articles | Metrics
    In this work, we adapt and compare implicity linear collocation method and iterated implicity linear collocation method for solving nonlinear two dimensional Fredholm integral equations of Hammerstein type using IMQ-RBFs on a non-rectangular domain. IMQs show to be the most promising RBFs for this kind of equations. The proposed methods are mesh-free and they are independent of the geometry of domain. Convergence analysis of the proposed methods together with some benchmark examples are provided which support their reliability and numerical stability.
    SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM
    Meng Huang, Zhiqiang Xu
    2020, 38(4):  638-660.  DOI: 10.4208/jcm.1902-m2018-0109
    Asbtract ( 66 )   PDF  
    References | Related Articles | Metrics
    We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank r matrix X ∈ Rn×r from m scalar measurements y#em/em#=a#em/em#?XX?a#em/em#, a#em/em# ∈ Rn, #em/em#=1, …, m. Such problem arises in a variety of applications such as quadratic regression and quantum state tomography. We present a novel algorithm, which is termed exponential-type gradient descent algorithm, to minimize a non-convex objective function f(U)=1/4m Σ#em/em#=1m (y#em/em#-a#em/em#?UU?a#em/em#)2. This algorithm starts with a careful initialization, and then refines this initial guess by iteratively applying exponential-type gradient descent. Particularly, we can obtain a good initial guess of X as long as the number of Gaussian random measurements is O(nr), and our iteration algorithm can converge linearly to the true X (up to an orthogonal matrix) with m=O (nr log(cr)) Gaussian random measurements.
    ON NEW STRATEGIES TO CONTROL THE ACCURACY OF WENO ALGORITHM CLOSE TO DISCONTINUITIES II: CELL AVERAGES AND MULTIRESOLUTION
    Sergio Amat, Juan Ruiz, Chi-Wang Shu
    2020, 38(4):  661-682.  DOI: 10.4208/jcm.1903-m2019-0125
    Asbtract ( 77 )   PDF  
    References | Related Articles | Metrics
    This paper is the second part of the article and is devoted to the construction and analysis of new non-linear optimal weights for WENO interpolation capable of rising the order of accuracy close to discontinuities for data discretized in the cell averages. Thus, now we are interested in analyze the capabilities of the new algorithm when working with functions belonging to the subspace L1L2 and that, consequently, are piecewise smooth and can present jump discontinuities. The new non-linear optimal weights are redesigned in a way that leads to optimal theoretical accuracy close to the discontinuities and at smooth zones. We will present the new algorithm for the approximation case and we will analyze its accuracy. Then we will explain how to use the new algorithm in multiresolution applications for univariate and bivariate functions. The numerical results confirm the theoretical proofs presented.