Finite Element Approximation of the Transport of Reactive Solutes in Porous Media. Part 1: Error Estimates for Nonequilibrium Adsorption Processes
John W. Barrett, Peter Knabner · SIAM Journal on Numerical Analysis · 1997
In this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration, backward Euler time discretization, and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media: find $\{u(x,t),v(x,t)\}$ such that \begin{eqnarray*} &\partial_t u + \partial_t v - \Delta u =f \quad\mbox{in } \Omega \times(0,T] \qquad u=0\quad\mbox{on } \partial\Omega\times(0,T]&\\ &\partial_tv=k(\varphi(u)-v) \quad\mbox{in }\Omega\times(0,T]&\\ &u(\cdot\,,0)=g_1(\cdot)\quad v(\cdot\,,0) =g_2(\cdot)\quad\mbox{in } \Omega\subset \Real^d,\quad 1\le d\le 3& \end{eqnarray*} for given data $k\in \Real^+$, f, $g_1$, $g_2$ and a monotonically increasing $\varphi\in C^0(\Real)\cap C^1(-\infty,0]\cup(0,\infty)$ satisfying $\varphi(0)=0$, which is only locally Hölder continuous with exponent $p\in(0,1)$ at the origin, e.g., $\varphi(s)\equiv[s]_+^p$. This lack of Lipschitz continuity at the origin limits the regularity of the unique solution $\{u,v\}$ and leads to difficulties in the finite element error analysis.