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 ∂tu+ ∂t[φ(u)]−∆u = f in Ω× (0, T ], u = 0 on ∂Ω× (0, T ] u(·, 0) = g(·) in Ω for known data Ω ⊂ R, 1 ≤ d ≤ 3, f , g, and a monotonically increasing φ ∈ C0(R) ∩ C1(−∞, 0] ∪ (0,∞) satisfying φ(0) = 0, which is only locally Hölder continuous with exponent p ∈ (0, 1) at the origin; e.g., φ(s) ≡ [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.