Error estimates for a mixed finite element discretization of some degenerate parabolic equations

Abstract

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one ) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart-Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.

A study on iterative methods for solving Richards’ equation

“…Galerkin finite elements were used in [2,3,9,22,24,32], often together with mass lumping to ensure a maximum principle [8]. Locally mass conservative schemes for Richards' equation were proposed and analysed in [10,11] (finite volumes), in [16] (multipoint flux approximation) or [4,5,26,29,34,35] (mixed finite element method). The analysis is performed mostly by using the Kirchhoff transformation (which combines the two main non-linearities in one) [1,4,26,28,34] or, alternatively, by restricting the generality, e.g.…”

Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media

“…Notice that the convergence order is still at least 2. This is in spite of the theoretical estimates of order τ + h 2 , which are obtained even in a weaker norm, but for the Richards equation in the saturated/unsaturated flow regime (see [5,31,33,37]). This improved convergence for the flow leads to better results for the solute transport, similar to the theoretical estimates in Theorem 4.5.…”

“…The regularity assumption on ∂ t Θ is given u p in [31], resulting in a reduction of the convergence order to τ + h 2 . A more general situation is considered in [33], where only Hölder continuity is assumed for Θ(·). In this case the convergence result applies to all flow regimes.…”

“…in [11]. Due to their local mass conservation property, mixed finite element methods are a valuable discretization technique for problems involving flow in porous media [3,4,5,11,12,16,17,31,33,37,41,42].…”

Hysteresis models and gravity fingering in porous media

“…The numerical schemes are described in more detail in [8] and [7], respectively. For other numerical approaches and detailed numerical analysis we refer to [12] and the references therein. For the inverted hysteresis relation (1.6), we use a regularized function Ψ δ γ,τ : R → R, 1) where δ > 0 is a regularizing parameter.…”