Finite element approximation of transport of reactive solutes in porous media. Part 2: Error estimates for equilibrium adsorption processes.
- Barrett, John W.
- Knabner, Peter
2010 Mathematics Subject Classification
- 65M15 65M60 35K65 35R35 35K55 76S05
- Finite element approximation, error estimates, degenerate parabolic equation, energy norm estimates, flow in porous media
In this paper we analyse a fully practical piecewise linear finite element approximation; involving numerical integration, backward Euler time discretisation 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)] - Δu = f in Ω x (0,T]
u = 0 on ∂Ω x (0,T] u(•,0) = g (•) in Ω
for known data Ω ⊂ ℝd, 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 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.