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

Abstract

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.