Finite element approximation of transport of reactive solutes in porous media. Part I: Error estimates for non-equilibrium adsorption processes.

In this paper we analyse a fully practical piecewise linear finite element approximation; involving regularization, numerical integration and backward Euler time discretisation; 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 ∂tu + ∂tv - Δu = f in Ω x (0,T] u=0 on ∂ Ω x (0,T] ∂tv = k (φ(u)-v) in Ω x (0,T] u(•,0) = g1 (•) v(•,0) = g 2 (•) in Ω ⊂ ℝd, 1 ≤ d ≤ 3 for given data k ∈ + ℝ+, f, g1, g2 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,v} and leads to difficulties in the finite element error analysis. Nevertheless we arrive at error bounds which in some cases exhibit the full approximation power of the trial space