Approximation of a nonlinear degenerate parabolic equation via a linear relaxation scheme

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.