Linearized Fe approximations to a nonlinear gradient flow

Buyang Li, Weiwei Sun

Research output: Journal article publicationJournal articleAcademic researchpeer-review

9 Citations (Scopus)

Abstract

We study fully discrete linearized Galerkin finite element approximations to a nonlinear gradient flow, applications of which can be found in many areas. Due to the strong nonlinearity of the equation, existing analyses for implicit schemes require certain restrictions on the time step and no analysis has been explored for linearized schemes. This paper focuses on the unconditionally optimal L error estimate of a linearized scheme. The key to our analysis is an iterated sequence of time-discrete elliptic equations and a rigorous analysis of its solution. We prove the W1,∞boundedness of the solution of the time-discrete system and the corresponding finite element solution, based on a more precise estimate of elliptic PDEs in W2,2+∈1and H2+∈2and a physical feature of the gradient-dependent diffusion coefficient. Numerical examples are provided to support our theoretical analysis.
Original languageEnglish
Pages (from-to)2623-2646
Number of pages24
JournalSIAM Journal on Numerical Analysis
Volume52
Issue number6
DOIs
Publication statusPublished - 1 Jan 2014
Externally publishedYes

Keywords

  • Error estimate
  • Finite element
  • Gradient flow
  • Nonlinear diffusion
  • Stability

ASJC Scopus subject areas

  • Numerical Analysis

Cite this