Next Articles

A TWO-GRID METHOD FOR THE C0 INTERIOR PENALTY DISCRETIZATION OF THE MONGE-AMPERE EQUATION

Gerard Awanou1, Hengguang Li2, Eric Malitz3   

  1. 1. Department of Mathematics, Statistics, and Computer Science, M/C 24
    9. University of Illinois at Chicago, Chicago, IL 60607, USA;
    2. Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, MI 48202, USA;
    3. Department of Mathematics, DePaul University, Chicago, IL 60614, USA
  • Received:2018-03-26 Revised:2018-10-23 Online:2020-07-15 Published:2020-07-15
  • Supported by:

    Gerard Awanou was partially supported by NSF DMS grant # 1720276 and Hengguang Li by NSF DMS grant # 1819041 and by NSF China Grant 11628104.

Gerard Awanou, Hengguang Li, Eric Malitz. A TWO-GRID METHOD FOR THE C0 INTERIOR PENALTY DISCRETIZATION OF THE MONGE-AMPERE EQUATION[J]. Journal of Computational Mathematics, 2020, 38(4): 547-564.

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.

CLC Number: 

[1] G. Awanou, Pseudo transient continuation and time marching methods for Monge-Ampère type equations, Adv. Comput. Math., 41:4(2015), 907-935.

[2] G. Awanou, Standard finite elements for the numerical resolution of the elliptic Monge-Ampère equation:Aleksandrov solutions, ESAIM Math. Model. Numer. Anal., 51:2(2017), 707-725.

[3] G. Awanou and H. Li, Error analysis of a mixed finite element method for the Monge-Ampère equation, Int. J. Num. Analysis and Modeling, 11(2014), 745-761.

[4] J.H. Bramble, J.E. Pasciak and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp., 47:175(1986), 103-134.

[5] S.C. Brenner, T. Gudi, M. Neilan and L.Y. Sung, C0 penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp., 80:276(2011), 1979-1995.

[6] S.C. Brenner, M. Neilan, A. Reiser and L.Y. Sung, A C0 interior penalty method for a von Kármán plate, Numer. Math., 135:3(2017), 803-832.

[7] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics, Springer, New York, third edition, 2008.

[8] X. Feng, R. Glowinski and M. Neilan, Recent Developments in Numerical Methods for Fully Nonlinear Second Order Partial Differential Equations, SIAM Rev., 55:2(2013), 205-267.

[9] E. Malitz, Two-grid discretization for interior penalty and mixed finite element approximations of the elliptic Monge-Ampère equation, Ph.D. Dissertation, University of Illinois at Chicago. USA, 2019.

[10] M. Neilan, Quadratic finite element approximations of the Monge-Ampère equation, J. Sci. Comput., 54:1(2013), 200-226.

[11] M. Neilan, Finite element methods for fully nonlinear second order PDEs based on a discrete Hessian with applications to the Monge-Ampère equation, J. Comput. Appl. Math., 263(2014), 351-369.

[12] V. Thomée, J. Xu and N.Y. Zhang, Superconvergence of the gradient in piecewise linear finiteelement approximation to a parabolic problem, SIAM J. Numer. Anal., 26:3(1989), 553-573.

[13] J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33:5(1996), 1759-1777.
[1] Weifeng Zhang, Shuo Zhang. ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS [J]. Journal of Computational Mathematics, 2020, 38(4): 565-579.
[2] H. Laeli Dastjerdi, M. Nili Ahmadabadi. IMPLICITY LINEAR COLLOCATION METHOD AND ITERATED IMPLICITY LINEAR COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF HAMMERSTEIN FREDHOLM INTEGRAL EQUATIONS ON 2D IRREGULAR DOMAINS [J]. Journal of Computational Mathematics, 2020, 38(4): 624-637.
[3] Meng Huang, Zhiqiang Xu. SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM [J]. Journal of Computational Mathematics, 2020, 38(4): 638-660.
[4] Sergio Amat, Juan Ruiz, Chi-Wang Shu. ON NEW STRATEGIES TO CONTROL THE ACCURACY OF WENO ALGORITHM CLOSE TO DISCONTINUITIES II: CELL AVERAGES AND MULTIRESOLUTION [J]. Journal of Computational Mathematics, 2020, 38(4): 661-682.
[5] Mohammad Tanzil Hasan, Chuanju Xu. HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY [J]. Journal of Computational Mathematics, 2020, 38(4): 580-605.
[6] Qilong Zhai, Xiaozhe Hu, Ran Zhang. THE SHIFTED-INVERSE POWER WEAK GALERKIN METHOD FOR EIGENVALUE PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(4): 606-623.
[7] Juncai He, Lin Li, Jinchao Xu, Chunyue Zheng. RELU DEEP NEURAL NETWORKS AND LINEAR FINITE ELEMENTS [J]. Journal of Computational Mathematics, 2020, 38(3): 502-527.
[8] Chunmei Xie, Minfu Feng. A NEW STABILIZED FINITE ELEMENT METHOD FOR SOLVING TRANSIENT NAVIER-STOKES EQUATIONS WITH HIGH REYNOLDS NUMBER [J]. Journal of Computational Mathematics, 2020, 38(3): 395-416.
[9] Leiwu Zhang. A STOCHASTIC MOVING BALLS APPROXIMATION METHOD OVER A SMOOTH INEQUALITY CONSTRAINT [J]. Journal of Computational Mathematics, 2020, 38(3): 528-546.
[10] Weichao Kong, Jianjun Wang, Wendong Wang, Feng Zhang. ENHANCED BLOCK-SPARSE SIGNAL RECOVERY PERFORMANCE VIA TRUNCATED ?2/?1-2 MINIMIZATION [J]. Journal of Computational Mathematics, 2020, 38(3): 437-451.
[11] Jie Chen, Zhengkang He, Shuyu Sun, Shimin Guo, Zhangxin Chen. EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS [J]. Journal of Computational Mathematics, 2020, 38(3): 452-468.
[12] Li Cai, Ye Sun, Feifei Jing, Yiqiang Li, Xiaoqin Shen, Yufeng Nie. A FULLY DISCRETE IMPLICIT-EXPLICIT FINITE ELEMENT METHOD FOR SOLVING THE FITZHUGH-NAGUMO MODEL [J]. Journal of Computational Mathematics, 2020, 38(3): 469-486.
[13] Liying Zhang, Jing Wang, Weien Zhou, Landong Liu, Li Zhang. CONVERGENCE ANALYSIS OF PARAREAL ALGORITHM BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS [J]. Journal of Computational Mathematics, 2020, 38(3): 487-501.
[14] Yijun Zhong, Chongjun Li. PIECEWISE SPARSE RECOVERY VIA PIECEWISE INVERSE SCALE SPACE ALGORITHM WITH DELETION RULE [J]. Journal of Computational Mathematics, 2020, 38(2): 375-394.
[15] Maohua Ran, Chengjian Zhang. A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS [J]. Journal of Computational Mathematics, 2020, 38(2): 239-253.
Viewed
Full text


Abstract