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

doi:10.4208/jcm.1901-m2018-0039

Journal of Computational Mathematics ›› 2020, Vol. 38 ›› Issue (4): 547-564.DOI: 10.4208/jcm.1901-m2018-0039

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

Gerard Awanou¹, Hengguang Li², Eric Malitz³

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.

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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 W^1,∞ 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.

Key words: Two-grid discretization, Interior penalty method, Finite element, MongeAmpè, re

CLC Number:

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

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