𝐿¹–framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations

Chen, Gui-Qiang; Karlsen, Kenneth

doi:10.1090/S0002-9947-04-03689-X

$L^1$–framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations
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by Gui-Qiang Chen and Kenneth H. Karlsen PDF

Trans. Amer. Math. Soc. 358 (2006), 937-963 Request permission

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.

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