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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Sharp $ L^1$ a posteriori error analysis for nonlinear convection-diffusion problems


Authors: Zhiming Chen and Guanghua Ji
Journal: Math. Comp. 75 (2006), 43-71
MSC (2000): Primary 65N15, 65N30, 65N50
Published electronically: September 29, 2005
MathSciNet review: 2176389
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Abstract: We derive sharp $ L^\infty(L^1)$ a posteriori error estimates for initial boundary value problems of nonlinear convection-diffusion equations of the form

$\displaystyle \frac{\partial u}{\partial t}+\operatorname{div}f(u)-\Delta A(u)=g$      

under the nondegeneracy assumption $ A'(s)>0$ for any $ s\in$R. The problem displays both parabolic and hyperbolic behavior in a way that depends on the solution itself. It is discretized implicitly in time via the method of characteristic and in space via continuous piecewise linear finite elements. The analysis is based on the Kruzkov ``doubling of variables'' device and the recently introduced ``boundary layer sequence'' technique to derive the entropy error inequality on bounded domains. The derived a posteriori error estimators have the correct convergence order in the region where the solution is smooth and recover the standard a posteriori error estimators known for parabolic equations with strong diffusions.


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Additional Information

Zhiming Chen
Affiliation: LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Email: zmchen@lsec.cc.ac.cn

Guanghua Ji
Affiliation: Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Email: ghji@lsec.cc.ac.cn

DOI: http://dx.doi.org/10.1090/S0025-5718-05-01778-3
PII: S 0025-5718(05)01778-3
Received by editor(s): September 3, 2004
Received by editor(s) in revised form: September 29, 2004
Published electronically: September 29, 2005
Additional Notes: This author was supported in part by China NSF under grant 10025102 and by China MOST under grant G1999032802 and 2005CB321700. Part of the work was done when the first author was participating in the 2003 Programme Computational Challenges in Partial Differential Equations at the Isaac Newton Institute for Mathematical Sciences, Cambridge, United Kingdom
Article copyright: © Copyright 2005 American Mathematical Society