Advances in Differential Equations

Error estimate for approximate solutions of a nonlinear convection-diffusion problem

Robert Eymard, Thierry Gallouët, and Raphaèle Herbin

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This paper proves the estimate $\Vert{{u_\varepsilon}} - u \Vert_{L^1(Q_T)} \le C \varepsilon^{1/5}$, where, for all $\varepsilon>0$, ${{u_\varepsilon}}$ is the weak solution of $({{u_\varepsilon}})_t + {\hbox{div}} ({{\bf q}} \ f({{u_\varepsilon}}))- {\Delta}(\varphi({{u_\varepsilon}})+\varepsilon {{u_\varepsilon}}) = 0$ with initial and boundary conditions, $u$ is the entropy weak solution of $u_t + {\hbox{div}} ({{\bf q}} f(u))- {\Delta}(\varphi(u)) = 0$ with the same initial and boundary conditions, and $C>0$ does not depend on $\varepsilon$. The domain $\Omega$ is assumed to be regular and $T$ is a given positive value.

Article information

Adv. Differential Equations, Volume 7, Number 4 (2002), 419-440.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K65: Degenerate parabolic equations
Secondary: 35K57: Reaction-diffusion equations 65M15: Error bounds 76M25: Other numerical methods 76R99: None of the above, but in this section


Eymard, Robert; Gallouët, Thierry; Herbin, Raphaèle. Error estimate for approximate solutions of a nonlinear convection-diffusion problem. Adv. Differential Equations 7 (2002), no. 4, 419--440.

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