A unified framework for two-grid methods for a class of nonlinear problems

Dongho Kim, Eun-Jae Park, Boyoon Seo

Research output: Contribution to journalArticle

Abstract

In this article, we present a unified error analysis of two-grid methods for a class of nonlinear problems. We first study the two-grid method of Xu by recasting the methodology in the abstract framework of Brezzi, Rappaz, and Raviart (BRR) for approximation of branches of nonsingular solutions and derive a priori error estimates. Our convergence results indicate that the correct scaling between fine and coarse meshes is given by h= O(H2) for all the nonlinear problems which can be written in and applied to the BRR framework. Next, a correction step can be added to the two-grid algorithm, which allows the choice h= O(H3). On the other hand, the particular BRR framework with duality pairing, if it is applied to a semilinear problem, allows a higher order relation h= O(H4). Furthermore, even the choice h= O(H5) is possible with the correction step either on fine mesh or coarse mesh. In addition, elliptic problems with gradient nonlinearities and the Naiver–Stokes equations are considered to illustrate our unified theory. Finally, numerical experiments are conducted to confirm our theoretical findings. Numerical results indicate that the correction step used as a simple postprocessing enhances the solution accuracy, particularly for problems with layers.

Original languageEnglish
Article number45
JournalCalcolo
Volume55
Issue number4
DOIs
Publication statusPublished - 2018 Dec 1

Fingerprint

Two-grid Method
Error analysis
Nonlinear Problem
Mesh
A Priori Error Estimates
Order Relation
Experiments
Post-processing
Error Analysis
Semilinear
Pairing
Elliptic Problems
Convergence Results
Duality
Branch
Numerical Experiment
Scaling
Nonlinearity
Higher Order
Gradient

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Computational Mathematics

Cite this

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A unified framework for two-grid methods for a class of nonlinear problems. / Kim, Dongho; Park, Eun-Jae; Seo, Boyoon.

In: Calcolo, Vol. 55, No. 4, 45, 01.12.2018.

Research output: Contribution to journalArticle

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