Pointwise error estimates of a linearized difference scheme for strongly coupled fractional Ginzburg‐Landau equations

2019 ◽  
Vol 43 (2) ◽  
pp. 512-535 ◽  
Author(s):  
Kejia Pan ◽  
Xianlin Jin ◽  
Dongdong He
Keyword(s):  
Difference Scheme ◽  
Error Estimates ◽  
Ginzburg Landau ◽  
Strongly Coupled ◽  
Pointwise Error ◽  
Ginzburg Landau Equations ◽  
Pointwise Error Estimates

scholarly journals A fourth-order linearized difference scheme for the coupled space fractional Ginzburg–Landau equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yuan Xu ◽  
Jiali Zeng ◽  
Shuanggui Hu
Keyword(s):  
Difference Scheme ◽  
Fourth Order ◽  
Landau Equation ◽  
Richardson Extrapolation ◽  
Implicit Difference Scheme ◽  
Ginzburg Landau ◽  
Pointwise Error ◽  
Ginzburg Landau Equations ◽  
Pointwise Error Estimates ◽  
Theoretical Results

Abstract In this paper, the coupled space fractional Ginzburg–Landau equations are investigated numerically. A linearized semi-implicit difference scheme is proposed. The scheme is unconditionally stable, fourth-order accurate in space, and second-order accurate in time. The optimal pointwise error estimates, unique solvability, and unconditional stability are obtained. Moreover, Richardson extrapolation is exploited to improve the temporal accuracy to fourth order. Finally, numerical results are presented to confirm the theoretical results.

scholarly journals Two-scale method for the Monge–Ampère equation: pointwise error estimates

2018 ◽  
Vol 39 (3) ◽  
pp. 1085-1109 ◽  
Author(s):  
R H Nochetto ◽  
D Ntogkas ◽  
W Zhang
Keyword(s):  
Error Estimates ◽  
Viscosity Solution ◽  
Continuous Dependence ◽  
Convergence Rates ◽  
Classical Solutions ◽  
Discrete Version ◽  
Scale Method ◽  
Pointwise Error ◽  
Pointwise Error Estimates ◽  
Monge Ampere

Abstract In this paper we continue the analysis of the two-scale method for the Monge–Ampère equation for dimension d ≥ 2 introduced in the study by Nochetto et al. (2017, Two-scale method for the Monge–Ampère equation: convergence to the viscosity solution. Math. Comput., in press). We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete version of the Alexandroff estimate. They are both instrumental to prove pointwise error estimates for classical solutions with Hölder and Sobolev regularity. We also derive convergence rates for viscosity solutions with bounded Hessians which may be piecewise smooth or degenerate.

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