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Volume 38, Issue 4
A Two-Grid Method for the C0 Interior Penalty Discretization of the Monge-Ampère Equation

Gerard Awanou, Hengguang Li & Eric Malitz

DOI: 10.4208/jcm.1901-m2018-0039

J. Comp. Math., 38 (2020), pp. 547-564.

Published online: 2020-04

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  • Abstract

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 $C^0$ 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 $W^{1,\infty}$ 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.


  • Keywords

Two-grid discretization, Interior penalty method, Finite element, Monge-Ampère.

  • AMS Subject Headings

65N30, 65N55, 35J96

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

li@wayne.edu (Hengguang Li)

EMALITZ@depaul.edu (Eric Malitz)

  • BibTex
  • RIS
  • TXT
@Article{JCM-38-547, author = {Awanou , GerardLi , Hengguang and Malitz , Eric}, title = {A Two-Grid Method for the C0 Interior Penalty Discretization of the Monge-Ampère Equation}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {4}, pages = {547--564}, abstract = {

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 $C^0$ 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 $W^{1,\infty}$ 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.


}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1901-m2018-0039}, url = {http://global-sci.org/intro/article_detail/jcm/16462.html} }
TY - JOUR T1 - A Two-Grid Method for the C0 Interior Penalty Discretization of the Monge-Ampère Equation AU - Awanou , Gerard AU - Li , Hengguang AU - Malitz , Eric JO - Journal of Computational Mathematics VL - 4 SP - 547 EP - 564 PY - 2020 DA - 2020/04 SN - 38 DO - http://doi.org/10.4208/jcm.1901-m2018-0039 UR - https://global-sci.org/intro/article_detail/jcm/16462.html KW - Two-grid discretization, Interior penalty method, Finite element, Monge-Ampère. AB -

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 $C^0$ 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 $W^{1,\infty}$ 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.


Awanou , GerardLi , Hengguang and Malitz , Eric. (2020). A Two-Grid Method for the C0 Interior Penalty Discretization of the Monge-Ampère Equation. Journal of Computational Mathematics. 38 (4). 547-564. doi:10.4208/jcm.1901-m2018-0039
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