Paper Info
Title
Unconditionally Optimal Error Analysis of Crank-Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System.
Abstract
In this paper, we present unconditionally optimal error estimates of linearized Crank---Nicolson Galerkin finite element methods for a strongly nonlinear parabolic system in $$\\mathbb {R}^d\\ (d=2,3)$$Rd(d=2,3). However, all previous works required certain time-step conditions that were dependent on the spatial mesh size. In order to overcome several entitative difficulties caused by the strong nonlinearity of the system, the proof takes two steps. First, by using a temporal-spatial error splitting argument and a new technique, optimal $$L^2$$L2 error estimates of the numerical schemes can be obtained under the condition $$\\tau \\ge h$$źźh, where $$\\tau $$ź denotes the time-step size and h is the spatial mesh size. Second, we obtain the boundedness of numerical solutions by mathematical induction and inverse inequality when $$\\tau \\le h$$ź≤h. Then, optimal $$L^2$$L2 and $$H^1$$H1 error estimates are proved in a different way for such case. Numerical results are given to illustrate our theoretical analyses.
Year
DOI
Venue
2017
10.1007/s10915-017-0381-3
J. Sci. Comput.
Keywords
Field
DocType
Unconditionally optimal error analysis, Linearized Crank–Nicolson scheme, Galerkin FEMs, A strongly nonlinear parabolic system
Inverse,Mathematical optimization,Nonlinear system,Parabolic system,Mathematical analysis,Galerkin method,Mathematical induction,Finite element method,Crank–Nicolson method,Mathematics
Journal
Volume
Issue
ISSN
72
2
1573-7691
Citations 
PageRank 
References 
8
0.56
10
Authors
2
Name
Order
Citations
PageRank
Dongfang Li110615.34
Jilu Wang2242.43