Pointwise error estimates of the bilinear SDFEM on Shishkin meshes

2012 ◽  
Vol 29 (2) ◽  
pp. 422-440 ◽  
Author(s):  
Jin Zhang ◽  
Liquan Mei
Keyword(s):  
Error Estimates ◽  
Shishkin Meshes ◽  
Pointwise Error ◽  
Pointwise Error Estimates

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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