23
ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU 1,2 , IULIU SORIN POP 3 , AND SABINE ATTINGER 1,2 Abstract. In this paper we analyze an Euler implicit-mixed finite element scheme for a porous media solute transport model. The transporting flux is not assumed given, but obtained by solving numerically the Richards equation, a model for sub-surface fluid flow. We prove the convergence of the scheme by estimating the error in terms of the discretization parameters. In doing so we take into account the numerical error occurring in the approximation of the fluid flow. The paper is concluded by numerical experiments, which are in good agreement with the theoretical estimates. 1. Introduction. The need of accurate and efficient numerical schemes to solve reaction-convection-diffusion equations modelling transport in porous media is well recognized. Inappropriate numerical methods can lead to false predictions for the evolution of a contaminated site, as revealed e.g. 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]. Here, we present and analyze an Euler implicit-mixed finite element scheme (EI-MFE) that is based on the lowest order Raviart-Thomas (RT 0 ) elements for the equation t (Θ(ψ)c) −∇· (Dc Qc) = Θ(ψ)r(c) in J × Ω, (1.1) with c denoting the concentration of the solute, D the diffusion-dispersion coefficient and r(·) a reaction term. Here J = (0,T ] (0 <T< ) is the time interval, whereas Ω IR d (d 1) is the computational domain having a Lipschitz continuous boundary Γ. The initial and boundary conditions are c(t = 0) = c I in Ω, and c =0 on J × Γ. (1.2) Equation (1.1) models the transport of one component involving non-equilibrium reactions. This situation is considered for the ease of presentation, but the present results can be extended to the case of a multi-component reactive transport, as long as the reactive term r(·) remains Lipschitz continuous. A general model for the reactive transport of M mobile and N immobile species is presented in [34]. To extend the applicability of the present analysis, in Section 4.1 we include also equilibrium sorption effects (see [8, 9, 18]), which complicates the analysis. We mention in particular the case of a Freundlich type isotherm, when the equation becomes degenerate. The case with sorption is considered separately only for an easier understanding of the ideas. The water flux Q appearing in (1.1), as well as the water content Θ, are ob- tained by solving the Richards equations modeling sub-surface water flow, including unsaturated regions near the surface: t Θ(ψ) −∇· (K(Θ(ψ))(ψ + z )) = 0 in J × Ω. (1.3) 1 UFZ-Helmholtz Center for Environmental Research, Permoserstr. 15, D-04318 Leipzig, Ger- many ([email protected]). 2 University of Jena, W¨ ollnitzerstr. 7, D-07749, Jena, Germany 3 Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands ([email protected]). 1

ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

  • Upload
    others

  • View
    27

  • Download
    0

Embed Size (px)

Citation preview

Page 1: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT

SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS

MEDIA

FLORIN A. RADU1,2 , IULIU SORIN POP3 , AND SABINE ATTINGER1,2

Abstract. In this paper we analyze an Euler implicit-mixed finite element scheme for a porousmedia solute transport model. The transporting flux is not assumed given, but obtained by solvingnumerically the Richards equation, a model for sub-surface fluid flow. We prove the convergence ofthe scheme by estimating the error in terms of the discretization parameters. In doing so we take intoaccount the numerical error occurring in the approximation of the fluid flow. The paper is concludedby numerical experiments, which are in good agreement with the theoretical estimates.

1. Introduction. The need of accurate and efficient numerical schemes to solvereaction-convection-diffusion equations modelling transport in porous media is wellrecognized. Inappropriate numerical methods can lead to false predictions for theevolution of a contaminated site, as revealed e.g. in [11]. Due to their local massconservation property, mixed finite element methods are a valuable discretizationtechnique for problems involving flow in porous media [3, 4, 5, 11, 12, 16, 17, 31,33, 37, 41, 42]. Here, we present and analyze an Euler implicit-mixed finite elementscheme (EI-MFE) that is based on the lowest order Raviart-Thomas (RT0) elementsfor the equation

∂t(Θ(ψ)c) −∇ · (D∇c− Qc) = Θ(ψ)r(c) in J × Ω,(1.1)

with c denoting the concentration of the solute, D the diffusion-dispersion coefficientand r(·) a reaction term. Here J = (0, T ] (0 < T < ∞) is the time interval, whereasΩ ⊂ IRd (d ≥ 1) is the computational domain having a Lipschitz continuous boundaryΓ. The initial and boundary conditions are

c(t = 0) = cI in Ω, and c = 0 on J × Γ.(1.2)

Equation (1.1) models the transport of one component involving non-equilibriumreactions. This situation is considered for the ease of presentation, but the presentresults can be extended to the case of a multi-component reactive transport, as long asthe reactive term r(·) remains Lipschitz continuous. A general model for the reactivetransport of M mobile and N immobile species is presented in [34]. To extend theapplicability of the present analysis, in Section 4.1 we include also equilibrium sorptioneffects (see [8, 9, 18]), which complicates the analysis. We mention in particular thecase of a Freundlich type isotherm, when the equation becomes degenerate. The casewith sorption is considered separately only for an easier understanding of the ideas.

The water flux Q appearing in (1.1), as well as the water content Θ, are ob-tained by solving the Richards equations modeling sub-surface water flow, includingunsaturated regions near the surface:

∂tΘ(ψ) −∇ · (K(Θ(ψ))∇(ψ + z)) = 0 in J × Ω.(1.3)

1UFZ-Helmholtz Center for Environmental Research, Permoserstr. 15, D-04318 Leipzig, Ger-many ([email protected]).

2University of Jena, Wollnitzerstr. 7, D-07749, Jena, Germany3Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O.

Box 513, 5600 MB Eindhoven, The Netherlands ([email protected]).

1

Page 2: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Here ψ denotes the pressure head, K the hydraulic conductivity and z the heightagainst the gravitational direction. The Richards equation results by using the Darcylaw

Q = −K(Θ(ψ))∇(ψ + z),(1.4)

in the mass balance equation for water, which is assumed incompressible

∂tΘ(ψ) + ∇ ·Q = 0.(1.5)

Equations (1.4) - (1.5) are completed by initial and boundary conditions

Ψ(t = 0) = ΨI in Ω, and Ψ = 0 on J × Γ.(1.6)

Notice the occurrence of the two functions, the soil-water retention curve Θ(ψ), asweel as the hydraulic conductivity K(Θ), for which several forms are proposed basedon laboratory experiments (see e.g. [13]). These functions are strictly increasingand bounded in unsaturated regions, where Θ is less than a maximal saturation ΘS .Furthermore, the functions are constant in saturated regions, where Θ = ΘS. For thefunctions that are commonly used, (1.3) is a degenerate elliptic-parabolic equation,and solving it numerically is a challenge in itself. However, in this paper we focus onthe solute transport and reaction. The underlying assumption is that these processesdo not influence the water flow, therefore Θ and Q are assumed known. In practicalcomputations we first solve (1.3) numerically, and then use the results in (1.1). Tobe specific, as for the solute transport equation (1.1), the numerical solution of theRichards equation is obtained by an EI-MFE scheme based on RT0 elements (see [30]for details). The scheme is briefly presented in Section 3 together with a review ofthe main results concerning the convergence for saturated/unsaturated flow.

In this paper we analyze the EI-MFE scheme for the transport equation (1.1), byemploying techniques that are similar to those used in [5, 31, 37]. The main resultshows the convergence of the fully discrete numerical scheme for (1.1). It is obtainedin a general framework, by taking into account the low regularity of the solution ofthe Richards equation. The order of convergence clearly depends on the accuracyof the scheme for water flow. We also show the existence and uniqueness for thesolution of the variational problems on which the numerical scheme is constructed.For algorithmic and implementation details we refer to [30, 34].

Several papers are considering numerical schemes for transport equations. Wemention [8, 9, 11] for a conformal FEM discretization, [22, 26] for finite volumeschemes, and [35, 36] for discontinuous Galerkin methods. Furthermore, a charac-teristic - mixed method is studied in [4], upwind MFEM are considered in [16, 17],whereas combined finite volume-mixed hybrid finite elements are employed in [20].Typically either a constant Θ is assumed, which corresponds to a saturated flow (see[8, 9, 16, 20]), or a Θ that does not depend on time (see [4, 35, 36]). Another possiblesimplification is to incorporate the term c∂tΘ in the reactive term r(·) and to assumethat the resulting rate remains Lipschitz continuous [11]. In the general case of asaturated-unsaturated flow this assumption is not satisfied since the factor ∂tΘ(ψ)needs not to be essentially bounded.

An interesting situation is considered in [17], where the term c∂tΘ is replacedby Ac, with A a positive constant. Error estimates are obtained without using theGronwall lemma. A similar situation appears in [20] where the divergence of the waterflux is assumed strictly negative: ∇ · Q = r ≤ 0. However, in a general saturated-unsaturated porous media flow such an assumption is not necessary true.

2

Page 3: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

In this paper we consider a MFE discretization of (1.1), requiring a divergenceform. Therefore we can not reduce the terms c∂tΘ and c∇ · Q by using (1.5), asusually done when dealing with conforming FE. This makes the analysis of a MFEdiscretization of (1.1) very challenging. Further, we do not assume that the divergenceof the water flux is constant or has a constant sign.

The paper is structured as follows. The following section introduces the nota-tions and the working assumptions. In Section 3, the numerical scheme to solve theRichards equation is presented and the main results regarding its convergence arereviewed. Next, Section 4 contains the numerical analysis for the solute transportmodel, including existence and uniqueness of a solution of the continuous and dis-crete variational problems. In a brief discussion ending this section we also includeequilibrium sorption effects, leading to a possible degeneracy in the solute equation.The analysis is completed by several experiments in Section 5, where the theoreticalestimates are confronted with the numerical results. In particular we have investigatedin how far the accuracy for the solute is influenced by the errors in approximating thefluid flow.

2. Notations and Assumptions. In what follows we make use of commonnotations in the functional analysis. By 〈·, ·〉 we mean the inner product on L2(Ω),or the duality pairing between H1

0 (Ω) and H−1(Ω). Further, ‖ · ‖ and ‖ · ‖1 stand forthe norms in L2(Ω) and H1(Ω), respectively. The functions in H(div; Ω) are vectorvalued, having a L2 divergence. By C we mean a positive constant, not depending onthe unknowns or the discretization parameters.

Furthermore, we let Th be a regular decomposition of Ω ⊂ Rd into closed d-

simplices; h stands for the mesh-size (see [15]). Here we assume Ω = ∪T∈ThT , hence

Ω is polygonal. In this way the errors caused by an approximation of a nonpolygonaldomain are avoided; we mention [25] for a detailed analysis. We will use the discretesubspaces Wh ⊂ L2(Ω) and Vh ⊂ H(div; Ω) defined as

Wh := p ∈ L2(Ω)| p is constant on each element T ∈ Th,

Vh := q ∈ H(div; Ω)| q|T = a + bx for all T ∈ Th.(2.1)

In other words, Wh denotes the space of piecewise constant functions, while Vh is theRT0 space (see [14]). Notice that ∇ · q ∈Wh for any q ∈ Vh.

We will use the following L2 projectors (see [14] and [29], p. 237):

Ph : L2(Ω) →Wh, 〈Phw − w,wh〉 = 0,(2.2)

respectively

Πh : H(div; Ω) → Vh, 〈∇ · (Πhv − v), wh〉 = 0,(2.3)

for all w ∈ L2(Ω),v ∈ H(div; Ω) and wh ∈Wh. For these operators we have

‖w − Phw‖ ≤ Ch‖w‖1, respectively ‖v − Πhv‖ ≤ Ch‖v‖1(2.4)

for any w ∈ H1(Ω) and v ∈ (H1(Ω))d.For the discretization in time we let N ∈ N be strictly positive, and define the

time step τ = T/N , as well as tn = nτ (n ∈ 1, 2, . . . , N). Given a function f definedon the interval J , we define:

fn

=1

τ

∫ tn

tn−1

f(t) dt, and fn = f(tn),(2.5)

3

Page 4: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

whenever n ∈ 1, . . . , N. For n = 0 we take f0

= f0 = f(0).Throughout this paper we make use of the following assumptions:

(A1) The rate function r : IR → IR is Lipschitz continuous with the constant Lr;furthermore, r(c) = 0 for all c ≤ 0.

(A2) The diffusion coefficient D does not depend on ψ or Q. For simplicity, letD = 1.

(A3) 1 ≥ ΘS ≥ Θ(x) ≥ ΘR > 0, ∀x ∈ IR.(A4) The initial cI is essentially bounded and positive; furthermore, ΨI ∈ L2(Ω).(A5) Q ∈ L∞(J × Ω) ∩ L2(J ;H1(Ω)).Remark 2.1. For the ease of presentation we take D = 1. The extension to a

positive definite tensor is immediate, requiring a minor change in Lemma 4.35 below.Remark 2.2. In (A3) we assume that Θ(·) is uniformly bounded by a strictly

positive constant. Since Θ stands for the water content, the boundedness of Θ(·)is a reasonable assumption. However, by taking the lower bound strictly positive wedisregard the case of a completely dry (fully unsaturated) medium. For commonly usedporous media models (see e.g. [13]) such an assumption holds, for example, in the caseof a homogeneous medium if the initial and boundary saturation (where prescribed)also exceed the lower limit. Furthermore, (A5) also implies that ∂tΘ(ψ) ∈ L2(J ×Ω).Since Θ(ψ) is essentially bounded, we immediately obtain Θ(ψ) ∈ C([0, T ];L2(Ω)).

Remark 2.3. (A5) is also assumed in recent papers referring to the discretizationof porous media flow models (see also [11, 20]). Previous results are under strongerassumptions: a constant divergence of the flux ([35, 36, 17]), a constant sign for thewater flux ([16]), or a a constant water flux [8, 9, 18].

3. Error estimates for Richards’ equation. In this section we review someconvergence results for the EI-MFE (RT0) discretization of the Richards equation.The particular choice of the finite element space is dictated by the lacking regularityof the solution. We consider three cases: the saturated flow regime (where Θ = ΘS),the strictly unsaturated flow regime (where Θ < ΘS), and the saturated-unsaturatedflow regime where both cases mentioned above are allowed. Denoting by ez theconstant gravitational vector and given the initial pressure ψI ∈ L2(Ω), we can definethe mixed, time integrated variational form of (1.5)–(1.4).

Problem 3.1. Find (ψ,Q) ∈ L2(J ;L2(Ω))×L2(J ; (L2(Ω))d) such that∫ t

0Q(s) ds ∈

L2(J ;H(div; Ω)), and

〈Θ(ψ(t)) − Θ(ψI), w〉 + 〈∇ ·∫ t

0

Q(s)ds, w〉 = 0,(3.1)

〈∫ t

0

K−1(Θ(ψ(s))Q(s)ds,v〉 − 〈∫ t

0

ψ(s)ds,∇ · v〉 + 〈∫ t

0

ez dt,v〉 = 0,(3.2)

for all t ∈ J , w ∈ L2(Ω) and v ∈ H(div; Ω).We mention [2, 27] for the existence and uniqueness of a solution for Problem 3.1.

In particular, in [2] it is shown that Θ(ψ) ∈ L∞(J ;L1(Ω)). By (A3) we also haveΘ(ψ) ∈ L∞((0, T )× Ω).

With n ∈ 1, 2, . . . , N we can define the fully discrete problem at time level n:Problem 3.2. Let ψn−1

h be given. Find (ψnh ,Q

nh) ∈Wh × Vh such that

〈Θ(ψnh) − Θ(ψn−1

h ), wh〉 + τ〈∇ · Qnh, wh〉 = 0,(3.3)

〈K−1(Θ(ψnh)Qn

h,vh〉 − 〈ψnh ,∇ · vh〉 + 〈ez,vh〉 = 0,(3.4)

4

Page 5: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

for all wh ∈Wh and vh ∈ Vh.The discrete initial pressure is taken such that Θ(ψ0

h) = PhΘ(ψI). Since ψ0h ∈ Wh is

piecewise constant, the same holds for Θ(ψ0h). Moreover, we have Θ(ψ0

h) ≥ ΘR > 0.Assuming that the flow is fully saturated, when Θ = ΘS for all times and almost

everywhere in Ω, the Richards equation becomes elliptic. In this case one obtains [14]Theorem 3.1. Assume that the flow is fully saturated. Then we have

‖ψ − ψh‖2 + ‖Q− Qh‖2 ≤ Ch2(3.5)

If the flow is strictly partially saturated (i.e. ΘR ≤ Θ < ΘS), then the Richardsequation is nonlinear but regular (non-degenerate). Then we have ([3])

Theorem 3.2. Let (ψ,Q) be the solution of Problem 3.1 and (ψnh ,Q

nh) ∈ Wh×Vh

(n ∈ 1, 2, . . . , N) solving the discrete problems 3.2. Assuming (A3), there holds

N∑

n=1

τ‖ψ(tn) − ψnh‖2 +

N∑

n=1

∫ tn

tn−1

‖Θ(ψ(t)) − Θ(ψnh)‖2 dt

+

N∑

n=1

τ‖Qn − Qnh‖2 +

N∑

n=1

τ‖Q(tn) − Qnh‖2 ≤ C(τ2 + h2).

(3.6)

The convergence order of the scheme numerical scheme (3.3) - (3.4) in the partiallysaturated case has been investigated numerically in several papers (see [11, 12], as wellas Section 5). To our knowledge, no rigorous error estimates have been proven up tonow in this specific case. This is due to the lacking regularity of the solution, as wellas the nonlinearities involved in the coefficient functions. An alternative approachis to combine the two nonlinearities in (1.3) into a single one and use the Kirchhofftransformation [2, 5, 31, 37]. For this approach, assuming that the saturation isa Lipschitz continuous function of the pressure, as well as ∂tΘ ∈ L2(J ;L2(Ω)), anτ2 + h2 convergence order is obtained in [5, 31, 37]. The regularity assumption on∂tΘ is given u p in [31], resulting in a reduction of the convergence order to τ +h2. Amore general situation is considered in [33], where only Holder continuity is assumedfor Θ(·). In this case the convergence result applies to all flow regimes. We also referto [41] for the analysis of an expanded mixed finite element scheme for equation (1.3).In the same spirit we mention another well recognized mass conservative method, themultipoint flux approximation method (MPFA) (see e.g. [6]). As proven in [21], thismethod can be successfully employed for solving the Richards equation.

In this paper we aim at proving the convergence of the numerical scheme for thesolute transport. The error estimates proven in the main result here are given interms of discretization parameters and depend also on the accuracy of the scheme forthe water flow. Therefore it is not so relevant which method was used to solve theRichards equation, as long as it produces convergent approximations for the waterflux and the saturation.

Having in mind (A5), we assume something similar for the discrete fluxes:(A5′) Qn

h ∈ L∞(Ω) for all n ∈ 1, . . . , N.4. Error estimates for the EI-MFE scheme for the solute transport

equation. In this section we prove the error estimates for the EI-MFE discretiza-tion of the solute transport equation. We first set a continuous mixed variationalformulations of (1.1) and analyse the existence and uniqueness of the solution. Wecontinue by giving some stability estimates for the solution of the continuous problemand present the fully discrete scheme. After showing the existence and uniqueness of

5

Page 6: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

a solution also in the discrete case we prove the convergence of the scheme. Followingthe discussion in the introduction, we assume that the pressure ψ and the water fluxQ, as well as their approximations ψn

h , Qnh, are known and satisfying the estimates in

Section 3.Problem 4.1. (The continuous problem) Find c ∈ L2(J ;L2(Ω)) and q ∈

L2(J ;H(div; Ω)) such that for almost every t ∈ J there holds

〈Θ(ψ(t))c(t) − Θ(ψI)cI , w〉 + 〈∇ ·∫ t

0

q ds, w〉 = 〈∫ t

0

Θ(ψ)r(c) ds, w〉,(4.1)

〈q(t),v〉 − 〈c(t),∇ · v〉 − 〈c(t)Q(t),v〉 = 0,(4.2)

for all w ∈ L2(Ω) and v ∈ H(div; Ω).Theorem 4.1. Assuming (A1)-(A5), the Problem 4.1 has an unique solution.Proof. To prove the existence of a solution we use the Richards equation (1.5)

and obtain

Θ(ψ)∂tc− ∆c+ Q · ∇c = Θ(ψ)r(c) in J × Ω.(4.3)

With boundary and initial conditions as in (1.2) and since D = 1, (1.1) and (4.3) areformally equivalent. Therefore we start with a solution to (4.3) and then construct asolution to the mixed formulation of (1.1). The existence and uniqueness of a solutionfor the transformed equation (4.3) is provided first in H1(J ;H−1(Ω))∩L2(J ;H1

0 (Ω))(see [24], Chapter VI.4; alternatively one can prove the result as a limit of solutionsto linearized problems and using the results in Chapter III). Moreover, by (A1) and(A4) the solution c remains essentially bounded and positive. Furthermore, using(A5), the regularity of Ω and since the boundary conditions are homogeneous we canimprove the regularity of the solution to c ∈ H1(J ;L2(Ω)) ∩ L2(J ;H2(Ω)).

Let now ζ ∈ H10 (Ω) fixed arbitrary. Since ∂tc ∈ L2(J × Ω), using the embedding

H1 → L4, for any ϕ ∈ L2(J ;H10 (Ω)) we have

∫ T

0

〈ζ∂tc, ϕ〉dt∣

∣ ≤∫ T

0

‖∂tc‖ ‖ζϕ‖dt ≤ C‖ϕ‖L2(J;H1(Ω)),

showing that ζ∂tc ∈ L2(J ;H−1(Ω)). Furthermore, since c is essentially bounded andin L2(J ;H2(Ω)), we also have ζc ∈ L2(J ;H1(Ω)). By the regularity of Θ(Ψ), foralmost all 0 ≤ s < t ≤ T we have

〈Θ(Ψ(t)), ζc(t)〉− 〈Θ(Ψ(s)), ζc(s)〉 =

∫ t

s

〈∂τΘ(Ψ(τ)), ζc(τ)〉+ 〈∂τ (ζc(τ)),Θ(Ψ(τ))〉dτ.

Using the Richards equation (1.5), this gives

〈Θ(Ψ(t)), ζc(t)〉 − 〈Θ(ΨI), ζcI〉 +

∫ t

0

〈∇ ·Q, ζc(τ)〉 − 〈∂τ (ζc(τ)),Θ(Ψ(τ))〉dτ = 0.

Since c is a weak solution of (4.3), we immediately obtain

〈Θ(Ψ(t))c(t), ζ〉 − 〈cIΘ(ΨI), ζ〉 +∫ t

0 〈∇ ·(

Qc(τ))

, ζ〉dτ

+∫ t

0 〈∇c(τ)),∇ζ〉dτ =∫ t

0 〈Θ(Ψ(τ))r(c(τ)), ζ〉dτ.(4.4)

Equation (4.4) holds for any ζ ∈ H10 (Ω). However, since c ∈ L2(J ;H2(Ω)), we can

transform the inner product in the last term on the left into −〈∆c(τ), ζ〉. Then by

6

Page 7: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

density arguments, (4.4) holds for any test function ζ ∈ L2(Ω). Identifying nowq = cQ−∇c ∈ L2(J ;H(div; Ω)) we have found a solution to Problem 4.1.

For the uniqueness we assume that Problem 4.1 has two solutions (c1,q1), (c2,q2)in L2(J × Ω) × L2(J ;H(div; Ω)). With c = c1 − c2 and q = q1 − q2 it follows thatfor almost all t > 0 we have

〈Θ(ψ(t))c(t), w〉 + 〈∇ ·∫ t

0

q(s) ds, w〉 = 〈∫ t

0

Θ(ψ)(r(c1) − r(c2)) ds, w〉,

〈q(t),v〉 − 〈c(t),∇ · v〉 − 〈c(t)Q(t),v〉 = 0,

for all w ∈ L2(Ω), v ∈ H(div; Ω) and for almost all t > 0. Taking into above w = c(t)

and v =∫ t

0 q ds and adding the resulting gives

〈Θ(ψ(t))c(t), c(t)〉 + 〈q(t),

∫ t

0

q ds〉

= 〈c(t)Q(t),

∫ t

0

q ds〉 + 〈∫ t

0

Θ(ψ)(r(c1) − r(c2)) ds, c(t)〉.(4.5)

Integrating (4.5) in time from 0 to a t′ ≤ T , since Θ ≥ ΘR we obtain

ΘR

∫ t′

0

‖c‖2 dt+1

2

∫ t′

0

q dt

2

≤∫ t′

0

〈c(t)Q(t),

∫ t

0

q ds〉 dt+

∫ t′

0

〈∫ t

0

Θ(ψ)(r(c1) − r(c2)) ds, c(t)〉 dt.(4.6)

Since Q ∈ L∞(J × Ω) using the Cauchy-Schwarz inequality leads to

ΘR

4

∫ t′

0

‖c‖2 dt+1

2

∫ t′

0

q dt

2

≤ C2

ΘR

∫ t′

0

∫ t

0

q ds

2

dt+1

ΘR

∫ t′

0

∫ t

0

Θ(ψ)(r(c1) − r(c2)) ds

2

dt.

(4.7)Recalling that Θ ≤ ΘS, using the Lipschitz continuity of r(·) and the Cauchy-Schwarzinequality, the inequality (4.7) becomes

ΘR

4

∫ t′

0

‖c‖2 dt+1

2

∫ t′

0

q dt

2

≤ C2

ΘR

∫ t′

0

∫ t

0

q ds

2

dt+L2

RΘ2S

ΘRt′∫ t′

0

∫ t

0

‖c‖2 dt,

(4.8)

for all t′ ∈ [0, T ]. Since both solutions satisfy the same initial data, employing theGronwall lemma gives ‖c(t)‖ = 0, implying the uniqueness of c. Furthermore, since∫ t′

0q dt = 0 for all t′ ∈ [0, T ], (4.5) also gives q = q1 −q2 = 0, yielding uniqueness for

Problem 4.1.Remark 4.1. As follows from the above proof, the solution has more regu-

larity than required in the statement of Problem 4.1. To be specific, we have c ∈H1(J ;L2(Ω)) ∩ L2(J ;H2(Ω)) ∩ L∞(J × Ω), whereas q ∈ L∞(J × Ω). Furthermore,since Q ∈ L2(J ; (H1(Ω))d) we immediately obtain q ∈ L2(J ; (H1(Ω))d). Also noticethat the regularity of c and Θ(ψ) allows writing (4.1) for all t.

7

Page 8: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Recalling the notations in (2.5), the following stability estimates will be usedlater.

Proposition 4.2. For the solution of Problem 4.1 we have

N∑

n=1

τ‖qn‖21 ≤ C(4.9)

N∑

n=1

τ‖cn − cn‖2 ≤ Cτ2(4.10)

N∑

n=1

∫ tn

tn−1

‖c(t) − cn‖2dt ≤ Cτ2(4.11)

N∑

n=1

∫ tn

tn−1

‖c(t) − cn‖2dt ≤ Cτ2(4.12)

N∑

n=1

τ‖cn‖21 ≤ C(4.13)

Proof. Since q ∈ L2(J ; (H1(Ω))d), the estimate (4.9) follows straightforwardly:

N∑

n=1

τ‖qn‖21 =

N∑

n=1

τ

1

τ

∫ tn

tn−1

q dt

2

1

≤N∑

n=1

1

τ

(

∫ tn

tn−1

‖q‖1 dt

)2

≤ C.

For (4.10) we use ∂tc ∈ L2(J ;L2(Ω)):

N∑

n=1

τ‖cn − cn‖2 =

N∑

n=1

1

τ

Ω

(

∫ tn

tn−1

c(tn) − c(t) dt)2dx =

N∑

n=1

1

τ

Ω

(

∫ tn

tn−1

∫ tn

t

∂tc(s) ds dt)2dx

≤N∑

n=1

τ

Ω

∫ tn

tn−1

∫ tn

tn−1

(∂tc)2 ds dt dx ≤

N∑

n=1

τ2‖∂tc‖2L2(tn−1,tn;L2(Ω)) ≤ Cτ2.

The proof for (4.11) and (4.12) follows similarly, whereas (4.13) is a consequenceof c ∈ L2(J ;H1(Ω)).

We now proceed with the EI-MFE scheme for Problem 4.1.Problem 4.2. (The discrete problem) Let n ∈ 1, . . . , N, and Θ(ψn

h), Θ(ψn−1h ),

Qnh, as well as cn−1

h be given. Find (cnh ,qnh) ∈Wh × Vh such that

〈Θ(ψnh )cnh − Θ(ψn−1

h )cn−1h , wh〉 + τ〈∇ · qn

h , wh〉 = τ〈Θ(ψnh )r(cnh), wh〉,(4.14)

〈qnh ,vh〉 − 〈cnh,∇ · vh〉 − 〈cnhQn

h,vh〉 = 0,(4.15)

for all wh ∈Wh and vh ∈ Vh.

Initially we take c0h =Ph(Θ(ψI)cI)

Ph(Θ(ψI)). The particular form of the initial data is allowed

by the lower bound on Θ and will be used when proving Theorem 4.5 below.Theorem 4.3. Assuming (A1)-(A5′) and that the time step τ is small enough,

the Problem 4.2 has an unique solution.Proof. We fist show uniqueness. Let (cnh,1,q

nh,1) and (cnh,2,q

nh,2) be two solutions

of Problem 4.2. With cnh := cnh,1 − cnh,2 ∈ Wh and qnh = qn

h,1 − qnh,2 ∈ Vh we have

〈Θ(ψnh)cnh , wh〉 + τ〈∇ · qn

h , wh〉 = τ〈Θ(ψnh )(r(cnh,1) − r(cnh,2), wh〉,(4.16)

8

Page 9: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

and

〈qnh ,vh〉 − 〈cnh,∇ · vh〉 − 〈cnhQn

h,vh〉 = 0,(4.17)

for all wh ∈Wh and vh ∈ Vh. Taking wh = cnh and vh = τqnh , adding the above gives

〈Θ(ψnh)cnh , c

nh〉 + τ ‖qn

h‖2

= τ〈cnhQnh,q

nh〉 + τ〈Θ(ψn

h )(r(cnh,1) − r(cnh,2)), cnh〉.

(4.18)

Using now (A1), (A3), (A5′) and the Cauchy-Schwarz inequality we immediately get

‖cnh‖2 + τ ‖qnh‖

2 ≤ Cτ‖cnh‖2 + Cτ2 ‖qnh‖

2.(4.19)

If τ is sufficiently small, this gives gives cnh = 0 and qnh = 0, so the solution is unique.

To show the existence of a solution for the fully discrete Problem 4.2 we letw1, . . . , wn1 ∪ v1, . . . ,vn2 be a basis for Wh × Vh and introduce the mappingP : R

n1+n2 → Rn1+n2 that will be defined below. To do so we first notice that given

α, α ∈ Rn1 and β, β ∈ R

n2 , with ξ = (α, β), ξ = (α, β) one can consider

((ξ, ξ)) := (α, α)n1 + τ(β, β)n2 , and ‖|ξ|‖ := ((ξ, ξ))1/2

for defining an inner product, respectively a norm on Rn1+n2 . Here (·, ·)p stands for

the euclidian inner product in Rp.

Further, any ξ = (α, β) ∈ Rn1+n2 determines uniquely a pair (w, v) ∈ Wh × Vh

by w =∑n1

k=1 αkwk, respectively v =∑n2

k=1 βkvk. Then we let ξ = (α, β) ∈ Rn1+n2

be given by

αk = 〈Θ(ψnh )w − Θ(ψn−1

h )cn−1h , wk〉 + τ〈∇ · v, wk〉 − τ〈Θ(ψn

h )r(w), wk〉

for all k = 1, . . . , n1, respectively

βk = 〈v,vk〉 − 〈w,∇ · vk〉 − 〈wQnh,vk〉

for all k = 1, . . . , n2. Having determined the above we define P(ξ) = ξ. Notice thatfinding a ξ such that P(ξ) = 0 immediately gives a solution to Problem 4.2.

Clearly, P is continuous. Moreover, for any ξ = (α, β) ∈ Rn1+n2 , with (w, v) ∈

Wh × Vh introduced above we have

((P(ξ), ξ)) = 〈Θ(ψnh )w−Θ(ψn−1

h )cn−1h , w〉+ τ‖v‖2− τ〈wQn

h,vk〉− τ〈Θ(ψnh )r(w), wk〉.

Recalling (A1) - (A5), using the Cauchy inequality, as well as the inequality of meanswe obtain

((P(ξ), ξ)) ≥ ΘR‖w‖2 + τ2‖v‖2 − τCQ

2 ‖w‖2

−(

τΘSLr + ΘR

2

)

‖w‖2 − 12ΘR

‖Θ(ψn−1h )cn−1

h ‖2,

where CQ = ‖Q‖L∞ and Lr is the Lipschitz constant of r. For τ small enough wehave ΘR − τ(CQ + 2LrΘS) ≥ m > 0, yielding

((P(ξ), ξ)) ≥ 1

2minm, 1

(

‖|ξ|‖2 −K)

,

9

Page 10: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

where K = ‖Θ(ψn−1h )cn−1

h ‖2/(2ΘR). This gives ((P(ξ), ξ)) ≥ minm, 1K > 0 for allξ ∈ R

n1+n2 satisfying ‖|ξ|‖2 = 2K, yielding the existence of a solution (see Lemma1.4, p. 140 in [39]).

In the fully saturated case, since Θ(ψ) = ΘS is constant, standard techniquesfor parabolic equations (see, e.g, [23]) can be used to prove the convergence of thescheme. In this case we obtain:

Theorem 4.4. (Fully saturated flow) Let (c,q) solve Problem 4.1, and (cnh,qnh)

solve the problems 4.2 for all n ∈ 1, . . . , N. Assuming (A1)-(A5), there holds

maxn=1,...,N

‖c(tn) − cnh‖2 +

N∑

n=1

τ‖q(tn) − qnh‖2 ≤ C(τ2 + h2).(4.20)

In the saturated-unsaturated flow regime we start with the following

Theorem 4.5. (Saturated-unsaturated flow) Let (c,q) solve Problem 4.1 and(cnh,q

nh) solve the problems 4.2 for all n ∈ 1, . . . , N. Assuming (A1)-(A5) we have

N∑

n=1

τ‖cn − cnh‖2 + τ2

N∑

n=1

(Πhqn − qn

h)

2

≤ C

τ2 + h2 +

N∑

n=1

τ‖Qn − Qnh‖2

+

N∑

n=1

τ‖Θ(ψ(tn)) − Θ(ψnh )‖2 +

N∑

n=1

∫ tn

tn−1

‖Θ(ψ) − Θ(ψnh)‖2 dt,

(4.21)

where the constant C does not depend on the discretization parameters.

Proof. Taking t = tn in (4.1) and recalling the notation in (2.5) we have

〈Θ(ψn)cn, w〉 + 〈n∑

k=1

τ∇ · qk, w〉 = 〈Θ(ψI)cI , w〉 + 〈n∑

k=1

τΘ(ψ)r(c)k, w〉,(4.22)

for all w ∈ L2(Ω).

Before dealing with the flux equation (4.2) we mention that the analysis in [5,37] carried out for the Richards equation leaves the flux equation unchanged, andconsiders the time integrated variant of the balance equation (4.1). Here we proceedas in [33] and integrate also (4.2) in time to obtain

〈qn,v〉 − 〈cn,∇ · v〉 − 〈cQn,v〉 = 0,(4.23)

for all v ∈ H(div; Ω). Summing up (4.14) written for the time steps tk, k = 1, . . . , n,subtracting the result from (4.22), as well as (4.15) from (4.23), and employing theprojectors defined in (2.2), (2.3) gives

〈Θ(ψn)cn−Θ(ψnh)cnh , wh〉+

n∑

k=1

τ〈∇·Πhqk−qk

h, wh〉 =

n∑

k=1

τ〈Θ(ψ)r(c)k−Θ(ψk

h)r(ckh), wh〉,

(4.24)for all wh ∈Wh and

〈qn − qnh ,vh〉 − 〈Phc

n − cnh,∇ · vh〉 − 〈cQn − Qnhc

nh,vh〉 = 0,(4.25)

for all vh ∈ Vh. The specific choice of the initial data gives no error at t = 0 in (4.24).

10

Page 11: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Defining

enq =

n∑

k=1

(Πhqk − qk

h), ∀n ∈ 1, . . . , N,(4.26)

we take wh = Phcn−cnh ∈ Wh in (4.24) and vh = τen

q ∈ Vh in (4.25), add the resultingand obtain

〈Θ(ψn)cn − Θ(ψnh )cnh, Phc

n − cnh〉 + τ〈qn − qnh, e

nq 〉 − τ〈cQn − Qn

hcnh, e

nq 〉

=

n∑

k=1

τ〈Θ(ψ)r(c)k − Θ(ψk

h)r(ckh), Phcn − cnh〉.

Summing the above for n = 1 to N yields

N∑

n=1

〈Θ(ψn)cn − Θ(ψnh)cnh, Phc

n − cnh〉 +

N∑

n=1

τ〈qn − qnh, e

nq 〉

=

N∑

n=1

τ〈cQn − Qnhc

nh, e

nq 〉 +

N∑

n=1

n∑

k=1

τ〈Θ(ψ)r(c)k − Θ(ψk

h)r(ckh), Phcn − cnh〉.

(4.27)

Further we estimate the terms in (4.27), which are denoted by T1, . . . , T4. T1 gives

T1 =

N∑

n=1

〈Θ(ψn)cn − Θ(ψnh)cnh , Phc

n − cnh〉

=

N∑

n=1

〈Θ(ψnh)(cn − cnh), Phc

n − cnh〉 +

N∑

n=1

〈(Θ(ψn) − Θ(ψnh))cn, Phc

n − cnh〉

=

N∑

n=1

〈Θ(ψnh)(cn − cn), cn − cnh〉 +

N∑

n=1

〈Θ(ψnh )(cn − cn), Phc

n − cn〉

+N∑

n=1

〈Θ(ψnh )(cn − cnh), cn − cnh〉 +

N∑

n=1

〈Θ(ψnh)(cn − cnh), Phc

n − cn〉

+

N∑

n=1

〈(Θ(ψn) − Θ(ψnh))cn, Phc

n − cn〉 +

N∑

n=1

〈(Θ(ψn) − Θ(ψnh))cn, cn − cnh〉

=: T11 + T12 + T13 + T14 + T15 + T16.(4.28)

Using (A3) and the Cauchy-Schwarz inequality, for any δ11 > 0 we have

T11 ≤ ΘS

2δ11

N∑

n=1

‖cn − cn‖2+

ΘSδ112

N∑

n=1

‖cn − cnh‖2.(4.29)

The estimate for T12 is similar:

T12 ≤ ΘS

2

N∑

n=1

‖cn − cn‖2 +ΘS

2

N∑

n=1

‖Phcn − cn‖2 .(4.30)

The term T13 in (4.28) is positive. There holds

T13 ≥ ΘR

N∑

n=1

‖cn − cnh‖2 .(4.31)

11

Page 12: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Furthermore, for any δ14 > 0 we have

T14 ≤ ΘSδ142

N∑

n=1

‖cn − cnh‖2 +ΘS

2δ14

N∑

n=1

‖Phcn − cn‖2.(4.32)

Similarly, using the boundedness of c (see Remark 4.1) we have

T15 ≤ CN∑

n=1

‖Θ(ψn) − Θ(ψnh)‖2 + C

N∑

n=1

‖Phcn − cn‖2 ,(4.33)

whereas for any δ16 > 0, T16 gives

T16 ≤ C

2δ16

N∑

n=1

‖Θ(ψn) − Θ(ψnh)‖2

+Cδ16

2

N∑

n=1

‖cn − cnh‖2.(4.34)

To estimate T2 we use the following elementary equality:

2N∑

n=1

〈an,n∑

k=1

ak〉 =

N∑

n=1

an

2

+N∑

n=1

‖an‖2 ,(4.35)

for any set of d-dimensional real vectors ak ∈ Rd (k ∈ 1, . . . , N, d ≥ 1). This gives

T2 =

N∑

n=1

τ〈qn − Πhqn, en

q 〉 +

N∑

n=1

τ〈Πhqn − qn

h , enq 〉

=N∑

n=1

τ〈qn − Πhqn, en

q 〉 +τ

2‖eN

q ‖2 +τ

2

N∑

n=1

‖Πhqn − qn

h‖2.(4.36)

To conclude with T2, we estimate the first term on the right, which is denoted by T21:

T21 ≤ 1

2

N∑

n=1

‖qn − Πhqn‖2 +

1

2

N∑

n=1

τ2‖enq ‖2.(4.37)

The convective term T3 is split into four terms, denoted T31, . . . , T34:

T3 =

N∑

n=1

τ〈cQn − Qncn, en

q 〉 +

N∑

n=1

τ〈Qncn − Qn

hcn, en

q 〉

+N∑

n=1

τ〈Qnh(cn − cn), en

q 〉 +N∑

n=1

τ〈Qnh(cn − cnh), en

q 〉.(4.38)

Using (A5) and the Cauchy-Schwarz inequality, for T31 we obtain

T31 ≤ 1

2

N∑

n=1

‖ 1

τ

∫ tn

tn−1

Q(t)(c(t) − cn)dt‖2 +τ2

2

N∑

n=1

‖enq ‖2

≤ C

N∑

n=1

∫ tn

tn−1

‖c(t) − cn‖2dt+τ2

2

N∑

n=1

‖enq ‖2.(4.39)

12

Page 13: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Similarly, using (A5′), for T32 and T33 we get

T32 ≤ C

2

N∑

n=1

‖Qn − Qnh‖2 +

τ2

2

N∑

n=1

‖enq ‖2,(4.40)

and

T33 ≤ C

2

N∑

n=1

‖cn − cn‖2 +τ2

2

N∑

n=1

‖enq ‖2.(4.41)

Furthermore, using again (A5′), for any δ34 > 0 we obtain

T34 ≤ Cδ342

N∑

n=1

‖cn − cnh‖2 +τ2

2δ34

N∑

n=1

‖enq ‖2.(4.42)

In estimating the reaction term, for any δ4 > 0 we have

T4 ≤N∑

n=1

n∑

k=1

τ

2δ4‖Θ(ψ)r(c)

k − Θ(ψkh)r(ckh)‖2 +

N∑

n=1

n∑

k=1

τδ42

‖Phcn − cnh‖2

≤N∑

n=1

n∑

k=1

τ

2δ4‖Θ(ψ)r(c)

k − Θ(ψkh)r(ckh)‖2 +

N∑

n=1

Tδ42

‖Phcn − cnh‖2(4.43)

We denote the terms on the right by T41 and T42, and use the inequality ‖f + g‖2 ≤2(‖f‖2 + ‖g‖2) to obtain

T42 ≤N∑

n=1

Tδ4‖cn − cnh‖2 +

N∑

n=1

Tδ4‖Phcn − cn‖2.(4.44)

For T41 we obtain in a similar manner

T41 ≤N∑

n=1

n∑

k=1

τ

δ4‖Θ(ψ)r(c)

k − Θ(ψkh)r(c)

k‖2

+N∑

n=1

n∑

k=1

δ4‖Θ(ψk

h)(r(c)k − r(ck))‖2 +

N∑

n=1

n∑

k=1

δ4‖Θ(ψk

h)(r(ck) − r(ckh))‖2.(4.45)

We denote the terms in the right hand side above by T411, T412 and T413. Using theboundedness of c (see Remark 4.1), the Lipschitz continuity of r(·), as well as theCauchy-Schwarz inequality, T411 gives

T411 =

N∑

n=1

n∑

k=1

τ

δ4

Ω

(

1

τ

∫ tk

tk−1

(Θ(ψ) − Θ(ψkh))r(c) dt

)2

dx

≤ 1

δ4

N∑

n=1

n∑

k=1

∫ tk

tk−1

‖Θ(ψ) − Θ(ψkh)‖2 dt ≤ T

δ4τ

N∑

n=1

∫ tn

tn−1

‖Θ(ψ) − Θ(ψnh )‖2 dt .(4.46)

Using (A3) and the Lipschitz continuity of r(·), for the second term in (4.45) we have

T412 =

N∑

n=1

n∑

k=1

2

δ4τ‖∫ tk

tk−1

(r(c(t)) − r(ck))Θ(ψkh)dt‖2

≤ 2Θ2SL

2r

δ4

N∑

n=1

n∑

k=1

∫ tk

tk−1

‖c(t) − ck‖2dt ≤ C

δ4τ

N∑

n=1

∫ tn

tn−1

‖c(t) − cn‖2dt.(4.47)

13

Page 14: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Finally, we use (A1) and (A3) to estimate T413:

T413 ≤ C

δ4

N∑

n=1

τ

n∑

k=1

‖ck − ckh‖2.(4.48)

Choosing δ11, δ14, δ16, δ34 and δ4 properly and using (4.27)–(4.48), a C > 0 notdepending on h or τ exists such that

1

C

N∑

n=1

‖cn − cnh‖2 + τ‖eNq ‖2

≤N∑

n=1

‖Θ(ψ(tn)) − Θ(ψnh)‖2 +

N∑

n=1

1

τ

∫ tn

tn−1

‖Θ(ψ) − Θ(ψnh)‖2 dt

+

N∑

n=1

∥Q

n − Qnh

2

+

N∑

n=1

‖qn − Πhqn‖2 +

N∑

n=1

‖cn − Phcn‖2

+1

τ

N∑

n=1

∫ tn

tn−1

‖c(t) − cn‖2 +1

τ

N∑

n=1

∫ tn

tn−1

‖c(t) − cn‖2 +

N∑

n=1

‖cn − cn‖2

+

N∑

n=1

τ2‖enq ‖2 +

N∑

n=1

τ

n∑

k=1

‖ck − ckh‖2.

(4.49)

Using the projector estimates in (2.4), the regularity of c and q stated in Remark4.1, as well as Proposition 4.2, applying the discrete Gronwall lemma to (4.49) gives

N∑

n=1

‖cn − cnh‖2 + τ‖eNq ‖2 ≤ C

τ +h2

τ+

N∑

n=1

∥Q

n − Qnh

2

+

N∑

n=1

‖Θ(ψ(tn)) − Θ(ψnh)‖2 +

N∑

n=1

1

τ

∫ tn

tn−1

‖Θ(ψ) − Θ(ψnh )‖2 dt

.

(4.50)

Now (4.21) follows straightforwardly.Remark 4.2. Using the stability estimates in Proposition 4.2, (4.21) also implies:

N∑

n=1

∫ tn

tn−1

‖c(t) − cnh‖2 dt+ τ2

N∑

n=1

qn − qnh

2

≤ C

τ2 + h2 +

N∑

n=1

τ∥

∥Q

n − Qnh

2

+

N∑

n=1

τ‖Θ(ψ(tn)) − Θ(ψnh)‖2 +

N∑

n=1

∫ tn

tn−1

‖Θ(ψ) − Θ(ψnh)‖2 dt

.

(4.51)

As follows from (4.21) and (4.51), the estimates for the MFEM scheme appliedto the reactive transport equation depend on the errors in the approximation of theRichards equation. In what follows we consider the explicit convergence order, de-pending only on the discretization parameters. As presented in Theorem 4.4 optimalestimates can be obtained in the fully saturated case. Furthermore, in the strictlyunsaturated flow regime we can use Theorem 3.2 to obtain the following

Theorem 4.6. (strictly unsaturated flow) Let (c,q) solve Problem 4.1 and (cnh,qnh)

solve the problems 4.2 for n ∈ 1, . . . , N. Assuming (A1)–(A5), there holds

N∑

n=1

τ‖cn − cnh‖2 +

N∑

n=1

∫ tn

tn−1

q(t) − qnh dt

2

≤ C(

τ2 + h2)

,(4.52)

14

Page 15: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

where the constant C does not depend on the discretization parameters.

4.1. Error estimates for the case with sorption. In this section we addequilibrium sorption effects to the reactive flow model (1.1). The results are givenfor the general, unsaturated-saturated flow case. In the case of a fully saturated flow,sharper results can be obtained.

With s denoting the concentration of the adsorbed solute, the transport equationbecomes (see, e.g. [8, 9, 18] and the references therein)

∂t(Θ(ψ)c) + ρb∂ts−∇ · (D∇c− Qc) = Θ(ψ)r(c) in J × Ω,(4.53)

with ρb denoting the density of the soil, which is assumed constant. Depending onthe fastness of the adsorption process, we can speak about equilibrium process (ifadsorption is much faster than diffusion or transport), or non-equilibrium (when alltime scales are in balance). In this paper we restrict to equilibrium kinetics, whichleads to a degenerate parabolic model. In this case we have

s = φ(c),(4.54)

with φ denoting a sorption isotherm. Typical examples of φ are discussed in [30]:linear, Freundlich, Langmuir or Freundlich-Langmuir. From mathematical point ofview, the most interesting is the Freundlich isotherm

φ(c) = cα, for c ≥ 0, and with α ∈ (0, 1].(4.55)

For completeness we extend the function φ by 0 for all negative arguments. Noticethe singularity of the derivative at c = 0, therefore φ is not Lipschitz continuous. Inthe general setting we assume

(A6) The sorption isotherm φ is nondecreasing and Holder continuous with anexponent α ∈ (0, 1], i. e. |φ(a) − φ(b)| ≤ C|a− b|α ∀ a, b ∈ IR.

As for Problem 4.1, the continuous mixed variational formulation of (4.53) readsProblem 4.3. (The continuous problem) Find c ∈ L2(J ;L2(Ω)) and q ∈

L2(J ;H(div; Ω)) such that for almost all t ∈ J there holds

〈Θ(ψ(t))c(t) − Θ(ψI)cI , w〉 + ρb〈φ(c(t)) − φ(cI), w〉

+〈∇ ·∫ t

0

q ds, w〉 = 〈∫ t

0

Θ(ψ)r(c) ds, w〉,(4.56)

〈q,v〉 − 〈c,∇ · v〉 + 〈cQ,v〉 = 0,(4.57)

for all w ∈ L2(Ω) and v ∈ H(div; Ω).The sorption model (4.53) may degenerate for c = 0. Here we are interested in

the convergence of the numerical scheme, and we do not focus on questions concerningthe existence, uniqueness and the regularity of a solution. Therefore we assume thatProblem 4.3 has a unique solution, having the regularity mentioned in Remark 4.1.This allows us maintaining the working framework of the previous section.

Below we define the fully discrete scheme for Problem 4.3:Problem 4.4. Let n ∈ 1, . . . , N and Θ(ψn

h),Θ(ψn−1h ),Qn

h, cn−1h be given. Find

(cnh,qnh) ∈ Wh × Vh such that

〈Θ(ψnh)cnh − Θ(ψn−1

h )cn−1h , wh〉 + ρb〈φ(cnh) − φ(cn−1

h ), wh〉

+τ〈∇ · qnh, wh〉 = τ〈Θ(ψn

h )r(cnh), wh〉,(4.58)

〈qnh ,vh〉 − 〈cnh,∇ · vh〉 + 〈cnhQn

h,v〉 = 0,(4.59)

15

Page 16: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

for all wh ∈Wh and vh ∈ Vh.Initially we take a c0h ∈ Wh such that Θ(ψ0

h)c0h + φ(c0h) = Ph(Θ(ψI)cI + φ(cI)). Thischoice, allowed since Θ(ψ0

h) ≥ ΘR > 0 while φ is nondecreasing, gives for any wh ∈ Wh

〈Θ(ψ0h)c0h + φ(c0h), wh〉 = 〈Θ(ψI)cI + φ(cI)), wh〉.(4.60)

For the discretization of Problem 4.3 we have the following convergence result:Theorem 4.7. (saturated-unsaturated flow) Let (c,q) solve Problem 4.3 and

(cnh,qnh) the solution of Problem 4.4 for all n ∈ 1, . . . , N. Assuming (A1)–(A6), a

C > 0 not depending on the discretization parameters exists such that

N∑

n=1

τ‖cn − cnh‖2 + τ2

N∑

n=1

(Πhqn − qn

h)

2

≤ C

τ4α

1 + α + h1+α +

N∑

n=1

τ‖Qn − Qnh‖2

+

N∑

n=1

τ‖Θ(ψ(tn)) − Θ(ψnh)‖2 +

N∑

n=1

∫ tn

tn−1

‖Θ(ψ) − Θ(ψnh)‖2 dt

.(4.61)

Proof. We follow the ideas in the proof of (4.21). When compared to the schemefor Problem 4.1, to discretize the case with sorption one adds the term ρb〈φ(cnh) −φ(cn−1

h ), wh〉 in (4.58). In the present proof we focus on the differences brought bythis additional term. Specifically, in the left hand side of (4.27) we have to add

TS = ρb

N∑

n=1

〈φ(cn) − φ(cnh), Phcn − cnh〉.

To estimate TS we split it into

TS = ρb

N∑

n=1

〈φ(cn) − φ(cnh), cn − cnh〉 + ρb

N∑

n=1

〈φ(cn) − φ(cnh), Phcn − cn〉(4.62)

and denote the terms on the right by TS1 and TS2. By the monotonicity of the sorptionisotherm φ, TS1 is positive; moreover, (A6) gives

TS1 ≥ ρb ‖φ(cn) − φ(cnh)‖1+α

α

L1+α

α (Ω).(4.63)

For TS2 we use the Young inequality

ab ≤ ap

p+bq

qfor any a, b > 0 and p, q ∈ (1,∞) such that

1

p+

1

q= 1.

For any δ > 0, with a = δ|φ(cn) − φ(cnh)|, b = |Phcn−cn|δ and p = 1+α

α this gives

TS2 ≤ αρb

1 + αδ1 + αα

N∑

n=1

‖φ(cn) − φ(cnh)‖1+α

α

L1+α

α (Ω)

+ρb

1 + αδ−(1+α)

N∑

n=1

‖Phcn − cn‖1+α

L1+α(Ω) .

(4.64)

Denoting the two terms above by TS21 and TS22, we notice that choosing δ properlyallows compensating the first one by (4.63). Since α ∈ (0, 1], to estimate TS22 we use

16

Page 17: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

(2.4), the continuous embedding L2(Ω) ⊆ L1+α(Ω), the stability estimates (4.10) and(4.13), as well as the Young inequality

x1+α ≤ 1 − α

2τ2 +

1 + α

2(α−1)1+α x2,

(valid for all x > 0) and obtain

TS22 ≤ ρb

1 + α

N∑

n=1

21+α

‖Phcn − cn‖1+α + ‖cn − cn‖1+α

≤ Ch1+αN∑

n=1

(

1 − α

2+

1 + α

2‖cn‖2

1

)

+CN∑

n=1

(

1 − α

2τ2 +

1 + α

2(α−1)1+α ‖cn − cn‖2)

)

≤ C

(

h1+α

τ+ τ + τ1+ 2(α−1)

1+α

)

.(4.65)

Now we can use (4.62)-(4.65) and proceed as in the proof of (4.21) to obtain theestimates in (4.61).

5. Numerical Results. This section contains some numerical simulations forthe flow and contaminant transport. To be specific, the water content as well as theflux are not consider as given, but obtained by solving the Richards equation (1.5).Once these being determined, we determine the solute concentration by solving (1.1).To do so, we apply the EI-MFE schemes presented in Sections 3 and 4.

With Ω = (0, 1)2 and T = 1 (hence J = (0, 1]), for determining the saturationand the flux we solve

∂tΘ(ψ) − ∆ψ = f inJ × Ω(5.1)

in two situations, which are called later as Example 1 and Example 2. First we considera linear problem by taking Θ(ψ) = ψ, whereas Θ(ψ) = ψ2 in the second case. Theright hand side f is chosen such that the equation (5.1) admits the analytical solution

ψex(t, x, y) = 4 − 2x− 4tx(1 − x)y2(1 − y)2,(5.2)

yielding a saturation Θ and a flux Q satisfying (A3) and (A5). The initial andboundary conditions are also chosen accordingly: ψ(0, x, y) = 4 − 2x, as well asψ = ψex along J × ∂Ω.

Next we consider the solute transport equation

∂t(Θ(ψ)c) −∇ · (D∇c− Qc) = Θ(ψ)r(c) + g inJ × Ω(5.3)

with Θ(ψ) and Q provided by (5.1). We choose D = 10.0, whereas r(c) = c2.Furthermore, g and the initial and boundary conditions are chosen such that theequation (5.3) has the exact solution

cex(t, x, y) = 100(1 − exp(−t))y4(1 − y)4(1 − x) + 1.(5.4)

17

Page 18: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Table 5.1

Numerical results for the Richards equation, Example 1.

N τ = h ER γR

1 0.1 7.099274e-05 —2 0.05 1.559765e-05 2.193 0.025 3.472974e-06 2.174 0.0125 8.717040e-07 1.995 0.00625 2.167524e-07 2.01

All simulations are carried out with τ = h. We work on a uniform mesh, startingwith h = 0.1 and refine it successively by halving the grid size and the time step. Forthe Richards equation we compute the error:

ER :=

N∑

n=1

τ‖ψex(tn) − ψnh‖2 +

N∑

n=1

τ‖Qex(tn) − Qnh‖2,(5.5)

whereas for the transport equation we calculate the error for the concentration:

ET :=N∑

n=1

τ‖cnex − cnh‖2.(5.6)

To compare the numerical results with the theoretical estimates we have estimatedthe reduction order of two two successive errors, computed for two discretizationparameters - say h1 and h2. If the estimate for the error E is hγ for some γ > 0, thenthe ratio E1/E2 should behave asymptotically as (h1/h2)

γ . The tables presenting thenumerical results have a generic structure: a column for the discretization parameters,another one for the errors, and finally a column for the exponent γ estimated fromtwo successive calculations.

We start with Examples 1 and 2, which are regular parabolic problems. Thesecases can be assimilated to a strictly unsaturated flow. Following Theorem 3.2, ER

should behave at least as τ2 + h2. Since τ = h, asymptotically we should obtainγR = 2. The numerical results for the first two examples are included in Tables 5.1and 5.3.

Using the saturation and the flux calculated above, we solve (5.3). The numericalresults are presented in Tables 5.2 and 5.4. Following Theorem 4.6, the asymptoticorder for ET should be again γT = 2. Notice the good agreement of the numericalresults with the theoretical estimates in Theorems 3.2 and 4.6.

Table 5.2

Numerical results for the solute transport, Example 1.

N τ = h ET γT

1 0.1 2.844167e-05 —2 0.05 6.539999e-06 2.123 0.025 1.561840e-06 2.064 0.0125 4.067868e-07 1.945 0.00625 1.029391e-07 1.98

In the next example the Richards equation (5.1) is degenerate. This is due to the

18

Page 19: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Table 5.3

Numerical results for the Richards equation, Example 2.

N τ = h ER γR

1 0.1 6.844420e-05 —2 0.05 1.538689e-05 2.153 0.025 3.461777e-06 2.154 0.0125 8.710324e-07 1.995 0.00625 2.167251e-07 2.00

Table 5.4

Numerical results for the solute transport, Example 2.

N τ = h ET γT

1 0.1 2.845842e-05 —2 0.05 6.544955e-06 2.123 0.025 1.563020e-06 2.064 0.0125 4.071064e-07 1.945 0.00625 1.010405e-07 2.01

nonlinearity

Θ(ψ) :=

19

6, if ψ < 1,

−ψ3

3+

3ψ2

2− 2ψ + 4, if 1 ≤ ψ ≤ 2,

10

3, if 2 < ψ.

(5.7)

It is to see that Θ(·) is C1 and Lipschitz continuous, whereas Θ′ = 0 on R\(1, 2).Therefore (5.1) is parabolic whenever ψ ∈ (1, 2), and elliptic for the other values ofψ. The initial and boundary conditions, as well as the right hand side f(·) are chosensuch that (5.1) is solved by

ψex(t, x, y) = 64tx(1 − x)y(1 − y).(5.8)

With the pressure above we calculate the water content Θ and the water flux Q andconsider the solute transport equation (5.3) with homogeneous Dirichlet boundaryconditions and zero initial condition. As before we take r(c) = c2, and the source gsuch that

cex(t, x, y) = tx(1 − x)y(1 − y)(5.9)

is a solution of (5.3). Tables 5.5 and 5.6 are presenting the numerical results for theRichards equation, respectively the solute transport. Notice that the convergenceorder is still at least 2. This is in spite of the theoretical estimates of order τ + h2,which are obtained even in a weaker norm, but for the Richards equation in thesaturated/unsaturated flow regime (see [5, 31, 33, 37]). This improved convergencefor the flow leads to better results for the solute transport, similar to the theoreticalestimates in Theorem 4.5.

Having in mind the above results, the following question arises naturally: in howfar the estimates in (4.21) are reflecting a correct dependence of the solute error on

19

Page 20: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Table 5.5

Numerical results for the Richards equation, Example 3.

N τ = h ER γR

1 0.1 1.448942e-01 —2 0.05 1.123385e-02 3.693 0.025 1.750283e-03 2.684 0.0125 4.331334e-04 2.015 0.00625 1.080408e-04 2.00

the errors in the flux and saturation? In this sense we mention the following situation:given a stationary flow and its numerical approximation satisfying

N∑

n=1

τ‖Qnex − Qn

h‖2 ≤ Chβ1 ,(5.10)

is it possible to obtain a more accurate approximation of the solute, say an error oforder hβ2 with β2 > β1?

This situation appears in many practical situations, when one wants to know if itmakes sense to use higher order finite elements for the solute transport, in combinationwith a lower order finite element approximation of the flow (see also [11, 12]). Havingin mind this very important question, we have carried out the last numerical test, byconsidering a constant saturation Θ = 1 and flux Q = (1, 0) in the solute transportequation (5.3). In discretizing (5.3) we use an approximation of the flux Q given byQh = (1 + 10

√h, 0), yielding for the given T and Ω

ER =

N∑

n=1

τ‖Qnex − Qn

h‖2 = 100h.(5.11)

Further we maintain the framework of Example 3. Table 5.7 presents the error ET

given in (5.6), as well as the estimated order of convergence γT . It is worth noticingthat the present computations suggest a decrease of the accuracy down to the order1, identical to the order of ER. This suggests an answer to the question above, inthe sense that the approximation of the solute should be in balance with the one forthe flow. Nevertheless, as revealed in several numerical experiments carried out forproblems that are not dominated by convection, or at least where the errors in theapproximation of the convection are not dominating, the reduction in the convergenceorder only appears on very fine grids.

Table 5.6

Numerical results for the solute transport, Example 3.

N τ = h ET γT

1 0.1 1.862225e-05 —2 0.05 1.912738e-06 3.283 0.025 2.730755e-07 2.814 0.0125 5.605202e-08 2.285 0.00625 1.318858e-08 2.09

6. Conclusions. We have analyzed a numerical scheme for a porous media so-lute transport model, where the transporting flux is obtained by solving numerically

20

Page 21: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Table 5.7

Numerical results for the solute transport, Example 4.

N τ = h ET γT

1 0.1 1.204681e-05 —2 0.05 5.097287e-06 1.243 0.025 2.299894e-06 1.154 0.0125 1.086988e-06 1.085 0.00625 5.278606e-07 1.04

the Richards equation. The spatial discretization is mixed and based on the lowestorder Raviart - Thomas finite elements, whereas the time stepping is performed bythe Euler implicit method. We have proven the convergence of the scheme by es-timating the error in terms of the discretization parameters. In doing so we havetaken into account the numerical error occurring in the approximation of the fluidflow. Furthermore, we avoid some of the commonly made assumptions concerningthe boundedness of the time derivative of the saturation, or a strict sign of the waterflux or of its divergence. The numerical experiments agree with the estimates derivedtheoretically.

Acknowledgements. Part of the work of the first author was done duringhis stay at the Max Planck Institute for Mathematics in the Sciences in Leipzig.The work of the second author was supported by the Dutch government through thenational program BSIK: knowledge and research capacity, in the ICT project BRICKS(http://www.bsik-bricks.nl), theme MSV1.

REFERENCES

[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.[2] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z.

183 (1983), pp. 311-341.[3] T. Arbogast, M. Obeyesekere and M. F. Wheeler, Numerical methods for the simulation

of flow in root-soil systems, SIAM J. Num. Anal. 30 (1993), pp. 1677-1702.[4] T. Arbogast and M. F. Wheeler, A characteristics-mixed finite-element method for

advection-dominated transport problems, SIAM J. Numer. Anal. 33 (1995), pp. 402-424.[5] T. Arbogast, M. F. Wheeler and N. Y. Zhang, A nonlinear mixed finite element method

for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal.33 (1996), pp. 1669-1687.

[6] I. Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids,Comput. Geosci. 6 (2002), pp. 404-432.

[7] J. Baranger, J. F. Maitre and F. Oudin, Connection between finite volume and mixed finiteelement methods, M2AN Math. Model. Numer. Anal. 32 (1995), pp. 445-465.

[8] J. W. Barrett and P. Knabner, Finite Element Approximation of the Transport of Reac-tive Solutes in Porous Media. Part 1: Error Estimates for Nonequilibrium AdsorptionProcesses, SIAM J. Numer. Anal. 34 (1997), pp. 201-227

[9] J. W. Barrett and P. Knabner, Finite Element Approximation of The Transport of ReactiveSolutes in Porous Media. Part II: Error Estimates for Equilibrium Adsorption Processes,SIAM J. Numer. Anal. 34 (1997), pp. 455-479

[10] P. Bastian, K. Birken, K. Johanssen, S. Lang, N. Neuß, H. Rentz-Reichert and C.

Wieners, UG–a flexible toolbox for solving partial differential equations, Comput. Visualiz.Sci. 1 (1997), pp. 27-40.

[11] M. Bause and P. Knabner, Numerical simulation of contaminant biodegradation by higherorder methods and adaptive time stepping, Comput. Visualiz. Sci. 7 (2004), pp. 61-78.

[12] M. Bause, Higher and lowest order mixed finite element approximation of subsurface flow prob-lems with solutions of weak regularity, Advanced in Water Resources 31 (2007), pp. 370-382.

21

Page 22: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

[13] J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in PorousMedia, Kluwer Academic, Dordrecht, 1991.

[14] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, NewYork, 1991.

[15] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North–Holland, Amsterdam,1978.

[16] C. Dawson, Analysis of an upwind-mixed finite element method for nonlinear contaminanttransport equations, SIAM J. Numer. Anal. 35 (1998), pp. 1709-1724.

[17] C. Dawson and V. Aizinger, Upwind-mixed methods for transport equations, ComputationalGeosciences 3 (1999), pp. 93-110.

[18] C. J. van Duijn and P. Knabner, Solute transport in porous media with equilibrium andnonequilibrium multiple-site adsorption: travelling waves, J. Reine Angew. Math. 415(1991), pp. 1–49.

[19] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.[20] R. Eymard, D. Hilhorst and M. Vohralık, A combined finite volume-nonconforming/mixed-

hybrid finite element scheme for degenerate parabolic problems, Numer. Math. 105 (2006),pp. 73–131.

[21] R. A. Klausen, F. A. Radu and G. T. Eigestad, Convergence of MPFA on triangulationsand for Richards’ equation, International Journal for Numerical Methods in Fluids (2008),DOI:10.1002/fld.1787.

[22] R. Klofkorn, D. Kroner and M. Ohlberger, Local adaptive methods for convection domi-nated problems, Internat. J. Numer. Methods in Fluids 40 (2002), pp. 79-91.

[23] P. Knabner and L. Angermann, Numerical methods for elliptic and parabolic partial differ-ential equations, Springer Verlag, 2003.

[24] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, Linear and QuasilinearEquations of Parabolic Type, American Mathematical Society, Providence, Rhode Island,1968.

[25] R. H. Nochetto and C. Verdi, Approximation of degenerate parabolic problems using nu-merical integration, SIAM J. Numer. Anal. 25 (1988), pp. 784-814.

[26] M. Ohlberger and C. Rohde, Adaptive finite volume approximations of weakly coupled con-vection dominated problems, IMA J. Numer. Anal. 22 (2002), pp. 253-280.

[27] F. Otto, L1-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differ-ential Equations 131 (1996), pp. 20–38.

[28] I. S. Pop, F. A. Radu and P. Knabner, Mixed finite elements for the Richards’ equation:linearization procedure, J. Comput. and Appl. Math. 168 (2004), pp. 365-373.

[29] A. Quarteroni and A. Valli, Numerical approximations of partial differential equations,Springer-Verlag, 1994.

[30] F. A. Radu, Mixed finite element discretization of Richards’ equation: error analysis andapplication to realistic infiltration problems, PhD Thesis, University of Erlangen-Nurnberg,2004.

[31] F. A. Radu, I. S. Pop and P. Knabner, Order of convergence estimates for an Euler implicit,mixed finite element discretization of Richards’ equation, SIAM J. Numer. Anal. 42 (2004),pp. 1452-1478.

[32] F. A. Radu, I. S. Pop and P. Knabner, On the convergence of the Newton method for themixed finite element discretization of a class of degenerate parabolic equation, NumericalMathematics and Advanced Applications, A. Bermudez de Castro et al. (editors), Springer,pp. 1194-1200, 2006.

[33] F. A. Radu, I. S. Pop and P. Knabner, Error estimates for a mixed finite elementdiscretization of some degenerate parabolic equations, Numerische Mathematik (2008),DOI:10.1007/s00211-008-0139-9, pp. 1-27.

[34] F. A. Radu, M. Bause, A. Prechtel and S. Attinger, A mixed hybrid finite element dis-cretization scheme for reactive transport in porous media, submitted.

[35] B. Riviere and M. F. Wheeler, Discontinuous Galerkin methods for flow and transportproblems in porous media, Communications in numerical methods in engineering 18 (2002),pp. 63-68.

[36] B. Riviere and M. F. Wheeler, Non conforming methods for transport with nonlinear reac-tion, Contemporany mathematics 295 (2002), pp. 421-432.

[37] E. Schneid, P. Knabner and F. A. Radu, A priori error estimates for a mixed finite elementdiscretization of the Richards’ equation, Numerische Mathematik 98 (2004), pp. 353-370.

[38] B. Schweizer, Regularization of outflow problems in unsaturated porous media with dry re-gions, J. Differential Equations 237 (2007), pp. 278–306.

[39] R. Temam, Navier-Stokes equations: theory and numerical analysis, AMS Chelsea Publishing,

22

Page 23: ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT ... · ANALYSIS OF AN EULER IMPLICIT - MIXED FINITE ELEMENT SCHEME FOR REACTIVE SOLUTE TRANSPORT IN POROUS MEDIA FLORIN A. RADU

Providence, RI, 2001.[40] J. M. Thomas, Sur l’analyse numerique des methodes d’elements finis hybrides et mixtes,

These d’Etat, Universite Pierre & Marie Curie (Paris 6), 1977.[41] C. Woodward and C. Dawson, Analysis of expanded mixed finite element methods for a

nonlinear parabolic equation modeling flow into variably saturated porous media, SIAM J.Numer. Anal. 37 (2000), pp. 701-724.

[42] I. Yotov, A mixed finite element discretization on non–matching multiblock grids for a de-generate parabolic equation arizing in porous media flow, East–West J. Numer. Math. 5(1997), pp. 211-230.

23