$L^1$--Framework for Continuous Dependence and Error Estimates for Quasilinear Anisotropic Degenerate Parabolic Equations

Author: Gui-Qiang Chen and Kenneth Karlsen

Title: $L^1$--Framework for Continuous Dependence and Error Estimates for Quasilinear Anisotropic Degenerate Parabolic Equations

Abstract
We develop a general $L^1$--framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach. We apply our $L^1$--framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an $L^1$ error estimate for an upwind-central finite difference scheme.
This article has appeared in:
Transactions of the American Mathematical Society (TAMS) , Vol. ?? , page ?? (2004).
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Author Address
    
    Department of Mathematics
    Northwestern University
    Evanston, IL 60208-2730
    gqchen@math.nwu.edu
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