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Approximation of a nonlinear degenerate parabolic equation via a linear relaxation scheme
Author(s) -
Slodička Marián
Publication year - 2005
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20033
Subject(s) - mathematics , discretization , relaxation (psychology) , degenerate energy levels , mathematical analysis , nonlinear system , backward euler method , function (biology) , partial differential equation , parabolic partial differential equation , convergence (economics) , psychology , social psychology , physics , quantum mechanics , evolutionary biology , economics , biology , economic growth
We study a nonlinear degenerate parabolic equation of the type$$ \partial _{t}u + \partial _{t} [u]_{+}^{p}- \nabla \cdot (\nabla u-qu) = f \,\,\,\,\, \rm{in} [0, \it{T}] \times \rm{\Omega} $$ accompanied by an initial datum and mixed boundary conditions. The symbol [ · ] + denotes the usual cutoff function. The problem represents a model of a reactive solute transport in porous media. The exponent p fulfills p ∈ (0, 1). This limits the regularity of a solution and leads to inconveniences in the error analysis. We design a new robust linear numerical scheme for the time discretization. This is based on a suitable combination of the backward Euler method and a linear relaxation scheme. We prove the convergence of relaxation iterations on each time point t i . We derive the error estimates in suitable function spaces for all values of p ∈ (0, 1). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.