Finite Element Approximation of The Transport of Reactive Solutes in Porous Media. Part II: Error Estimates for Equilibrium Adsorption Processes

In this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration, backward Euler time discretization, and possibly regularization and relaxation of the following degenerate parabolic equation arising in a model of reactive solute transport in porous media: find $u(x,t)$ such that $$ \partial_t u + \partial_t [ \p (u)] - \Delta u = f \quad {\rm in} \quad \Omega \times (0,T\,], $$ \vspace*{-12pt} $$ u = 0 \quad {\rm on} \quad \partial \Omega \times (0,T\,] \qquad u(\cdot ,0) = g(\cdot ) \quad {\rm in} \; \Omega $$ for known data $ \Omega \subset {\bf R}^d, \; 1 \leq d \leq 3 $, f, g, and a monotonically increasing $ \p \in {\rm C}^0({\bf R}) \cap {\rm C}^1(- \infty , 0] \cup (0,\infty)$ satisfying $ \p (0) = 0$, which is only locally Hölder continuous with exponent $ p \in (0,1)$ at the origin; e.g., $ \p (s) \equiv [s]_+^p $. This lack of Lipschitz continuity at the origin limits the regularity of the unique solution u and leads to difficulties in the finite element error analysis.