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

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 \begin{eqnarray*} \frac {\partial u}{\partial t}+\operatorname {div}f(u)-\Delta A(u)=g \end{eqnarray*} under the nondegeneracy assumption $A’(s)>0$ for any $s\in \mathbb {R}$. 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 Kružkov “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.