Preprint 2001-008

An error estimate for viscous approximate solutions of degenerate parabolic equations

Abstract: Relying on recent advances in the theory of entropy solutions for nonlinear (strongly) degenerate parabolic equations, we present a direct proof of an L¹ error estimate for viscous approximate solutions of the initial value problem for $\pt w+\Div \bigl(V(x)f(w)\bigr) = \Delta A(w)$, where $V=V(x)$ is a vector field, $f=f(u)$ is a scalar function, and $A'(\cdot)\ge 0$. The viscous approximate solutions are weak solutions of the initial value problem for the uniformly parabolic equation $\pt w^{\eps}+\Div \bigl(V(x)f(w^{\eps})\bigr) = \Delta \bigl(A(w^{\eps}) +\eps w^{\eps}\bigr)$, $\eps>0$. The error estimate is of order $\sqrt{\eps}$.

Paper:

Available as PostScript (261 Kbytes) or gzipped PostScript (98 Kbytes; uncompress using gunzip).

Author(s):

Steinar Evje, <Steinar.Evje@rf.no>

Kenneth H. Karlsen, <kennethk@mi.uib.no>

Publishing information:

UCLA Computational and Applied Mathematics Report.

Comments:

Submitted by:

<kennethk@mi.uib.no> February 15 2001.