Eindhoven University of Technology · pore-scale simulation of crystal dissolution and precipitation in porous media in Free Boundary Problems: Theory and Applications, Vol. I (Irsee,

Convergence analysis of mixed numerical schemes forreactive flow in a porous mediumCitation for published version (APA):Kumar, K., Pop, I. S., & Radu, F. A. (2013). Convergence analysis of mixed numerical schemes for reactive flowin a porous medium. SIAM Journal on Numerical Analysis, 51(4), 2283-2308. https://doi.org/10.1137/120880938

DOI:10.1137/120880938

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SIAM J. NUMER. ANAL. c© 2013 Society for Industrial and Applied MathematicsVol. 51, No. 4, pp. 2283–2308

CONVERGENCE ANALYSIS OF MIXED NUMERICAL SCHEMESFOR REACTIVE FLOW IN A POROUS MEDIUM∗

K. KUMAR† , I. S. POP‡ , AND F. A. RADU§

Abstract. This paper deals with the numerical analysis of an upscaled model describing thereactive flow in a porous medium. The solutes are transported by advection and diffusion andundergo precipitation and dissolution. The reaction term and, in particular, the dissolution termhave a particular, multivalued character, which leads to stiff dissolution fronts. We consider the Eulerimplicit method for the temporal discretization and the mixed finite element for the discretization inspace. More precisely, we use the lowest order Raviart–Thomas elements. As an intermediate stepwe consider also a semidiscrete mixed variational formulation (continuous in space). We analyzethe numerical schemes and prove the convergence to the continuous formulation. Apart from theproof for the convergence, this also yields an existence proof for the solution of the model in mixedvariational formulation. Numerical experiments are performed to study the convergence behavior.

Key words. numerical analysis, reactive flows, weak formulation, implicit scheme, mixed finiteelement discretization

AMS subject classifications. 35A35, 65L60, 65J20

DOI. 10.1137/120880938

1. Introduction. Reactive flows in a porous medium have a wide range of ap-plications ranging from spreading of polluting chemicals leading to ground water con-tamination (see [40] and references therein) to biological applications such as tissueand bone formation, or pharmaceutical applications [27], or technological applicationssuch as operation of solid batteries. A common feature of the above applications isthe transport and reactions of ions/solutes. In this work, we deal with the transportof ions/solutes taking place through the combined process of convection and diffusion.For reactions, we focus on a specific class, namely, the precipitation and dissolutionprocesses, where the ions undergo combination (precipitation) to form a crystal. Thereverse process of dissolution takes place where the crystal gets dissolved.

We consider here an upscaled model defined on a Darcy scale. This implies thatthe solid grains and the pore space are not distinguished and the equations are definedeverywhere. Consequently, the crystals formed as a result of reactions among ions andthe ions themselves are defined everywhere in the domain. Such models fall in thegeneral category of reactive porous media flow models. For Darcy-scale models relateddirectly to precipitation and dissolution processes we refer to [7, 29, 33, 34] (see alsothe references therein). Here we adopt the ideas proposed first in [23], and extendedin a series of papers [15, 16, 17]. These papers are referring to Darcy-scale models; thepore-scale counterpart is considered in [18], where distinction is made for the domains

∗Received by the editors June 13, 2012; accepted for publication (in revised form) May 28, 2013;published electronically August 1, 2013. The authors are members of the International ResearchTraining Group NUPUS funded by the German Research Foundation DFG (GRK 1398) and by theNetherlands Organisation for Scientific Research NWO (DN 81-754).

http://www.siam.org/journals/sinum/51-4/88093.html†CASA, Technische Universiteit Eindhoven, Eindhoven 5600 MB, The Netherlands (k.kundan@

gmail.com). The work of this author was supported by STW project 07796.‡CASA, Technische Universiteit Eindhoven, Eindhoven 5600 MB, The Netherlands and Institute

of Mathematics, University of Bergen, Bergen 5008, Norway ([email protected]).§Institute of Mathematics, University of Bergen, Bergen 5008, Norway ([email protected].

no). This author’s work was partially supported by Akademia grant 2012-13 from Statoil.

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2284 K. KUMAR, I. S. POP, AND F. A. RADU

delineating the pore space and the solid grains. The transition from the pore-scalemodel to the upscaled model is obtained, for instance, via homogenization arguments.For a simplified situation of a two-dimensional (2D) strip, the rigorous arguments areprovided in [18]; see also [28, 1] for the upscaling procedure in transport dominatedflow regimes. For a similar situation, but tracking the geometry changes due to thereactions leading to the free boundary problems, the formal arguments are presentedin [24] and [30].

We are motivated by analyzing appropriate numerical methods for solving thereactive flows for an upscaled model. Considering the mixed variational formulationis an attractive proposition as it preserves the mass locally. Our main goal here isto provide the convergence of a mixed finite element discretization for such a modelfor dissolution and precipitation in porous media, involving a multivalued dissolutionrate. Before discussing the details and specifics, we briefly review some of the relevantnumerical works. For continuously differentiable rates the convergence of (adaptive)finite volume discretizations is studied in [22, 32]; see also [10] for the convergenceof a finite volume discretization of a copper-leaching model. In a similar framework,discontinuous Galerkin methods are discussed in [39] and upwind mixed finite elementmethods (MFEMs) are considered in [11, 12]; combined finite volume–mixed hybridfinite elements are employed in [20, 21]. Non-Lipschitz, but Holder continuous ratesare considered using conformal finite element method (FEM) schemes in [4, 5]. Sim-ilarly, for Holder continuous rates (including equilibrium and nonequilibrium cases),MFEMs are analyzed rigorously in [36, 38], whereas [37] provides error estimates forthe coupled system describing unsaturated flow and reactive transport. In all thesecases, the continuity of the reaction rates allows error estimates to be obtained. Acharacteristic MFEM for the advection dominated transport has been treated in [3]and characteristic FEM schemes for contaminant transport giving rise to possiblynon-Lipschitz reaction rates are treated in [13] where the convergence and the errorestimates have been provided. A parabolic problem coupled with linear ODEs at theboundary have been treated in [2] using a characteristic MFEM. Conformal schemesboth for the semidiscrete and fully discrete (FEM) cases for the upscaled model underconsideration have been treated in [25].

The main difficulty here is due to the particular description of the dissolution rate,which may become discontinuous. To deal with this, we consider a regularization ofthis term and the corresponding sequence of regularized equations. The regularizationparameter δ is dependent on the time discretization parameter τ in such a way thatas τ ↘ 0, it is ensured that δ ↘ 0. Thus, obtaining the limit of the discretizedscheme automatically yields, by virtue of the regularization parameter also vanishing,the original equation. In proving the convergence results, compactness arguments areemployed. These arguments rely on a priori estimates providing weak convergence.However, strong convergence is needed to deal with the nonlinear terms in the reactionrates. Translation estimates are used to achieve this.

We consider both the semidiscrete and the fully discrete cases with the prooffor the latter case following closely the ideas of the semidiscrete case. However,there are important differences, particularly in the way the translation estimates areobtained. Whereas in the semidiscrete case, we use the dual problem for obtaining thetranslation estimates, in the fully discrete case, we use the properties of the discreteH1

0 norm following the finite volume framework [19]. The convergence analysis ofappropriate numerical schemes for the problem considered here is a stepping stonefor coupled flow and reactive transport problems (for example, Richards’ equationcoupled with precipitation-dissolution reaction models).

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NUMERICAL ANALYSIS 2285

The paper is structured as follows. We begin with a brief description of themodel in section 2. We define the mixed variational formulation in section 2.2 wherewe prove the uniqueness of the solution with the existence coming from the conver-gence proof. Next, in sections 3 and 4 the time-discrete, respectively, fully discretenumerical schemes are considered and the proofs for the convergence are provided.The numerical experiments are shown in section 5 followed by the conclusions anddiscussions in section 6.

2. The mathematical model. We consider a Darcy-scale model that describesthe reactive transport of the ions/solutes in a porous medium. The solutes are sub-jected to convective transport and in addition they undergo diffusion and reactionsin the bulk. Below we provide a brief description and the assumptions of the model;we refer to [15], or [16] for more details.

Let Ω ⊂ R2 be the domain occupied by the porous medium, and assume Ω open,

connected, bounded, and with Lipschitz boundary Γ. Further, let T > 0 be a fixedbut arbitrarily chosen time, and define ΩT = (0, T ]× Ω, and ΓT = (0, T ]× Γ. At theoutset, we assume that the fluid velocity q is known, divergence free, and essentiallybounded.

Usually, two or more different types of ions react to produce precipitate (animmobile species). A simplified model will be considered here where we include onlyone mobile species. This makes sense if the boundary and initial data are compatible(see [15], or [16]). Then, denoting by v the concentration of the (immobile) precipitate,and by u the cation concentration, the model reduces to

(2.1)

⎧⎨⎩

∂t(u+ v) +∇ · (qu−∇u) = 0 in ΩT ,u = 0 on ΓT ,u = uI in Ω, for t = 0,

for the ion transport, and

(2.2)

⎧⎨⎩

∂tv = (r(u) − w) on ΩT ,w ∈ H(v) on ΩT ,v = vI on Ω, for t = 0,

for the precipitate. For the ease of presentation we restrict ourselves to homogeneousDirichlet boundary conditions. The assumptions for the initial conditions will begiven below.

In the system considered above, we assume all the quantities and variables asdimensionless. To simplify the exposition, the diffusion is assumed to be 1, the exten-sion to a positive definite diffusion tensor being straightforward. Further, we assumethat the Damkohler number is scaled to 1, as well as an eventual factor in the timederivative of v in (2.2)1, appearing in the transition form the pore scale to the corescale.

The assumptions on the precipitation rate r are the following:(Ar1) r(·) : R → [0,∞) is Lipschitz continuous in R with Lipschitz constant Lr;(Ar2) there exists a unique u∗ ≥ 0, such that

r(u) =

{0 for u ≤ u∗,strictly increasing for u ≥ u∗ with r(∞) = ∞.

(2.3)

The interesting part is the structure of the dissolution rate. We interpret it as aprocess encountered strictly at the surface of the precipitate layer, so the rate is

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assumed constant (1, by scaling) at some (t, x) ∈ ΩT where the precipitate is present,i.e., if v(t, x) > 0. In the absence of the precipitate, the overall rate (precipitate minusdissolution) is either zero, if the solute present there is insufficient to produce a netprecipitation gain, or positive. This can be summarized as

w ∈ H(v), where H(v) =

⎧⎨⎩

{0} if v < 0,[0, 1] if v = 0,{1} if v > 0.

(2.4)

In the setting above, a unique u∗ exists for which r(u∗) = 1. If u = u∗ for all tand x, then the system is in equilibrium: no precipitation or dissolution occurs, sincethe precipitation rate is balanced by the dissolution rate regardless of the presence ofabsence of crystals (see [25, section 5] for some illustrations). Then, as follows from[23, 18, 31], for a.e. (t, x) ∈ ΩT where v = 0, the dissolution rate satisfies

w =

{r(u) if u < u∗,

1 if u ≥ u∗.(2.5)

Since we will work with the model in the mixed formulation, we define the flux as

(2.6) Q = −∇u+ qu.

2.1. Notation. We adopt the standard notation from functional analysis. Inparticular, by H1

0 (Ω) we mean the space of functions in H1(Ω), having a vanishingtrace on Γ, and H−1 is its dual. By (·, ·) we mean the L2 inner product or the dualitypairing between H1

0 and H−1. Further, ‖ · ‖ stands for the norms induced by the L2

inner product. For other norms, we explicitly state it. The functions in H(div; Ω) arevector valued having an L2 divergence. Furthermore, C denotes a generic constantthe value of which might change from line to line and is independent of unknownvariables or the discretization parameters.

Having introduced this notation we can state the assumptions on the initial con-ditions:(AI1) The initial data uI and vI are nonnegative and essentially bounded;(AI2) uI , vI ∈ H1

0 (Ω).We have taken the initial conditions in H1

0 to avoid technicalities. Alternatively,one can approximate the initial conditions by taking the convolutions with smoothfunctions. The H1

0 regularity for vI is used for obtaining strong convergence results,for which L2 regularity is not sufficient.

We furthermore assume that Ω is polygonal. Therefore it admits regular de-compositions into simplices and the errors due to nonpolygonal domains are avoided.The spatial discretization will be defined on such a regular decomposition Th into 2Dsimplices (triangles); h stands for the mesh size. We provide the exposition for twodimensions but extending the results to three dimensions is similar. We define thefollowing sets:

V := H1((0, T );L2(Ω)),

S := L2((0, T );H(div; Ω)),

W := {w ∈ L∞(ΩT ); 0 ≤ w ≤ 1}.In addition, for the fully discrete situation, we use the following discrete subspaces

Vh ⊂ L2(Ω) and Sh ⊂ H(div; Ω) defined as follows:

Vh := {u ∈ L2(Ω) | u is constant on each element T ∈ Th},Sh := {Q ∈ H(div; Ω) | Q|T = a+ bx for all T ∈ Th}.

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In other words, Vh denotes the space of piecewise constant functions, while Sh is theRT0 space. Clearly from the above definitions, ∇ ·Q ∈ Vh for any Q ∈ Sh.We also define the following usual projections:

Ph : L2(Ω) �→ Vh, 〈Phv − v, vh〉 = 0

for all vh ∈ Vh. Similarly, the projection Πh is defined on (H1(Ω))2 such that

Πh : (H1(Ω))2 �→ Sh, 〈∇ · (ΠhQ−Q), vh〉 = 0

for all vh ∈ Vh. Following [35, p. 237] (also see [9]), this operator can be extended toH(div; Ω) and also for the above operators there holds

(2.7)

‖v − Phv‖ ≤ Ch‖v‖H1(Ω) for all v ∈ H1(Ω),

‖Q−ΠhQ‖ ≤ Ch‖Q‖H1(Ω) for all Q ∈ (H1(Ω))2,

‖∇ ·Q−∇ · (ΠhQ)‖ ≤ Ch‖Q‖H2(Ω) for all Q ∈ (H2(Ω))2.

For the spatial discretization we will work with the approximation qh of the Darcyvelocity q, defined on the given mesh Th. For this approximation we assume thatthere exists an Mq > 0 such that (s.t.) ‖qh‖L∞ ≤ Mq (the same estimate being validfor q), and as h ↘ 0

(2.8) ‖qh − q‖ → 0.

Having stated the assumptions, we proceed by introducing the mixed variationalformulation and analyzing the convergence of its discretization.

2.2. Continuous mixed variational formulation. Except for some particularsituations, one cannot expect the existence of classical solutions to (2.1)–(2.2) and wework with the weak formulation. A weak solution of (2.1)–(2.2) written in mixedform is defined as follows.

Definition 2.1. A quadruple (u,Q, v, w) ∈ V×S×V×W with u|t=0 = uI , v|t=0 =vI is a mixed weak solution of (2.1)–(2.2) if w ∈ H(v) a.e. and for all t ∈ (0, T ) and(φ, θ,ψ) ∈ L2(Ω)× L2(Ω)×H(div; Ω) we have

(2.9)(∂tu, φ) + (∇ ·Q, φ) + (∂tv, φ) = 0,

(∂tv, θ)− (r(u)− w, θ) = 0,(Q,ψ)− (u,∇ ·ψ)− (qu,ψ) = 0.

The proof for the existence of a solution for (2.9) is obtained by the convergenceof the numerical schemes considered below. Therefore, we give the proof for theuniqueness of the solution. The following lemma shows the uniqueness without furtherdetails on w. As mentioned in (2.5) the inclusion w ∈ H(v) can be made more precise.

Lemma 2.2. The mixed weak formulation (2.9) has at most one solution.Proof. Assume there exist two solution quadruples (u1,Q1, v1, w1) and (u2,Q2, v2,

w2), and define u := u1−u2, Q := Q1−Q2, v := v1−v2, w := w1−w2. Clearly,at t = 0 we have u(0, x) = 0 and v(0, x) = 0 for all x.

Subtracting (2.9)2 for u2, v2, and w2 from the equations for u1, v1, and w1 andtaking (for t ≤ T arbitrary) θ = χ(0,t)v, using monotonicity of H and the Lipschitzcontinuity of r(·) leads to

‖v(t, ·)‖2 =

∫ t

0

∫Ω

(r(u1)− r(u2))v(s, x)dxds −∫ t

0

∫Ω

(H(v1)−H(v2))v(s, x)dxds

≤ 1

2

∫ t

0

L2r‖u(s, ·)‖2ds+

1

2

∫ t

0

‖v(s, ·)‖2ds.

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Then Gronwall’s lemma gives

‖v(t, ·)‖2 ≤ C

∫ t

0

‖u(s, ·)‖2ds.(2.10)

Next, we choose for φ = χ(0,t)u(t, x) in the difference between the two equalities (2.9)1to get

‖u(t, ·)‖2 +(∫ t

0

∇ ·Q(s, ·)ds, u(t, ·))+ (v(t, ·), u(t, ·)) = 0.

Similarly, choosing ψ =

∫ t

0

Q(s)ds in (2.9)3 (written for a.e. t) yields

∫Ω

(Q(t, x)

∫ t

0

Q(s, x)ds

)dx−

∫Ω

u(t, x)

(∫ t

0

∇ ·Q(s, x)ds

)dx

=

∫Ω

qu(t, x)

(∫ t

0

Q(s, x)ds

)dx.

Combining the above gives

‖u(t, ·)‖2 +∫Ω

v(t, x)u(t, x)dx +

∫Ω

Q(t, x)

∫ t

0

Q(s, x)dsdx

≤ 1

4‖u(t, ·)‖2 +M2

q

∥∥∥∥∫ t

0

Q(s, ·)ds∥∥∥∥2

,

which implies

‖u(t, ·)‖2 + (Q(t, ·),∫ t

0

Q(s, ·)ds) ≤ 1

2‖u(t, ·)‖2 +M2

q

∥∥∥∥∫ t

0

Qds

∥∥∥∥2

+ ‖v(t, ·)‖2.

Using (2.10) we obtain

1

2‖u(t, ·)‖2 + (Q(t, ·),

∫ t

0

Q(s, ·)ds) ≤ C

∫ t

0

‖u(s, ·)‖2ds+M2q

∥∥∥∥∫ t

0

Q(s, ·)ds∥∥∥∥2

.

(2.11)

The uniqueness follows now by applying Gronwall’s lemma.

3. Semidiscrete mixed variational formulation. As announced, to avoiddealing with inclusion in the description of dissolution rate, the numerical schemerelies on the regularization of the Heaviside graph. With this aim, with δ > 0 wedefine

Hδ(z) :=

⎧⎨⎩

1, z > δ,zδ , 0 ≤ z ≤ δ,0, z < 0.

(3.1)

Next, with N ∈ N, τ = TN , and tn = nτ, n = 1, . . . , N , we consider a first order

time discretization with uniform time stepping, which is implicit in u and explicitin v. At each time step tn we use (un−1

δ , vn−1δ ) ∈

(L2(Ω), L2(Ω)

)determined at

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tn−1 to find the next approximation (unδ ,Q

nδ , v

nδ , w

nδ ). The procedure is initiated with

u0 = uI , v0 = vI . Specifically, we look for (un

δ , vnδ ,Q

nδ ) ∈ (L2(Ω), L2(Ω), H(div; Ω))

satisfying the following time-discrete problem.Problem Pmvf,n

δ . Given (un−1δ , vn−1

δ ) ∈(L2(Ω), L2(Ω)

), find (un

δ ,Qnδ , v

nδ , w

nδ ) ∈(

L2(Ω), H(div; Ω), L2(Ω), L∞(Ω))such that

(unδ − un−1

δ , φ) + τ(∇ ·Qnδ , φ) + (vnδ − vn−1

δ , φ) = 0,

(vnδ − vn−1δ , θ)− τ(r(un

δ ), θ)− τ(Hδ(vn−1δ ), θ) = 0,(3.2)

(Qnδ ,ψ)− (un

δ ,∇ · ψ)− (qunδ ,ψ) = 0

for all (φ, θ,ψ) ∈(L2(Ω), L2(Ω), H(div; Ω)

). For completeness we define wn

δ =

Hδ(vnδ ). This is a system of elliptic equations for un

δ ,Qnδ , v

nδ given un−1

δ ∈ H10,ΓD

(Ω),

vn−1δ ∈ L2(Ω). For stability reasons, we choose δ = O(τ

12 ) (see [14, 25] for detailed

arguments) which implies that τδ goes to 0 as τ ↘ 0. This in turn allows us to

consider the solutions along the sequence of regularized Heaviside function with theregularization parameter δ automatically vanishing in the limit of τ ↘ 0.

The existence of a solution for Problem Pmvf,nδ will result from the convergence

of the fully discrete scheme, which is proved in the appendix by keeping τ and δ fixed,and passing to the limit h ↘ 0. For now, we prove the uniqueness of the solution.

Lemma 3.1. Problem Pmvf,nδ has at most one solution triple (un

δ ,Qnδ , v

nδ ).

Proof. Since wnδ = Hδ(v

nδ ), it has no influence on the existence or uniqueness of

the solution. Therefore, we consider only the triples (unδ ,Q

nδ , v

nδ ). Assume that for

the same (un−1δ , vn−1

δ ) there are two solution triples (unδ,i,Q

nδ,i, v

nδ,i), i = 1, 2 providing

a solution to Problem Pmvf,nδ . Define

unδ := un

δ,1 − unδ,2, Qn

δ := Qnδ,1 −Qn

δ,2, vnδ := vnδ,1 − vnδ,2.

We now consider the equations for the differences above. Taking θ = vnδ in (3.2)2gives

‖vnδ ‖2 = τ(r(unδ,1)− r(un

δ,2), vnδ ) ≤ τLr‖un

δ ‖‖vnδ ‖

as the Hδ terms cancel because of explicit discretization. This gives, ‖vnδ ‖ ≤ Cτ‖unδ ‖.

Further, with φ = unδ , θ = un

δ ,ψ = τQnδ , from (3.2) we obtain

‖unδ ‖2 + τ‖Qn

δ ‖2 + τ(r(unδ,1)− r(un

δ,2), unδ ) = τ(qun

δ ,Qnδ ).

Since r is monotone, the Cauchy–Schwarz inequality and boundedness of q give

‖unδ ‖2 +

1

2τ‖Qn

δ ‖2 ≤ τ1

2M2

q ‖unδ ‖2.

For τ < 2M2

q, we obtain ‖un

δ ‖ = 0 and thereby ‖Qnδ ‖ = 0. Together with the above

bound on ‖vnδ ‖, we conclude unδ = vnδ = 0 and Qn

δ = 0.We start with the following stability estimates.Lemma 3.2. It holds that

supk=1,...,N

(∥∥ukδ

∥∥+ 1

τ

∥∥vkδ − vk−1δ

∥∥+ ∥∥vkδ ∥∥+ ∥∥∥Qkδ

∥∥∥) ≤ C,(3.3)

N∑n=1

(1

τ

∥∥unδ − un−1

δ

∥∥2 + ∥∥Qnδ −Qn−1

δ

∥∥2 + τ ‖∇ ·Qnδ ‖

2

)≤ C.(3.4)

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Proof. We start by showing (3.3). To this aim we choose φ = unδ , ψ =

τQnδ , θ = un

δ as test functions in (3.2), and add the resulting to obtain

(unδ − un−1

δ , unδ ) + τ ‖Qn

δ ‖2 − τ(qun

δ ,Qnδ ) + τ(r(un

δ ), unδ ) = τ(Hδ(v

n−1δ ), un

δ ).(3.5)

Since q and Hδ are bounded and r(unδ )u

nδ ≥ 0, by Young’s inequality we get

‖unδ ‖

2 −∥∥un−1

δ

∥∥2 + ∥∥unδ − un−1

δ

∥∥2 + 2τ ‖Qnδ ‖

2+ 2τ(r(un

δ ), unδ )

= 2τ(qunδ ,Q

nδ ) + 2τ(Hδ(v

n−1δ ), un

δ ) ≤ τ ‖Qnδ ‖

2 + Cτ ‖unδ ‖

2 + Cτ + Cτ ‖unδ ‖

2 .

Summing over n = 1, . . . , k (where k ∈ {1, . . . , N} is arbitrary) gives

∥∥ukδ

∥∥2 + k∑n=1


δ

∥∥2 + τ

k∑n=1

‖Qnδ ‖

2 ≤ ‖uI‖2 + C + Cτ

k∑n=1

‖unδ ‖

2,(3.6)

and the first term in (3.3) follows from the discrete Gronwall lemma.For the second term in (3.3), consider (3.2)2, choose θ = vnδ − vn−1

δ , and applythe Cauchy–Schwarz inequality for the right-hand side,

∥∥vnδ − vn−1δ

∥∥2 ≤ τ ‖r(unδ )‖


∥∥+ τ∥∥Hδ(v

n−1δ )

∥∥ ∥∥vnδ − vn−1δ

∥∥ .Using the previous bound, the boundedness of Hδ and the Lipschitz continuity of rimply the conclusion. To prove the third term, choose θ = vnδ in (3.2)2, rewrite theright-hand side, and using the monotonicity of Hδ and the Cauchy–Schwarz inequality

‖vnδ ‖2 −

∥∥vn−1δ

∥∥2 + ∥∥vnδ − vn−1δ

∥∥2 ≤ 2τC ‖unδ ‖ ‖vnδ ‖+ 2τ

(Hδ(v

n−1δ ), vnδ − vn−1

δ

).

By Young’s inequality this leads to

‖vnδ ‖2 −

∥∥vn−1δ

∥∥2 + 1

2


∥∥2 ≤ τ ‖vnδ ‖2 + Cτ ‖un

δ ‖2 + 2τ2 ‖Hδ‖2 .

Summing over n = 1, . . . , k (with k ∈ {1, . . . , N} arbitrary), this gives

∥∥vkδ ∥∥2 + k∑n=1


∥∥2 ≤ ‖vI‖2 + τ

k∑n=1

‖vnδ ‖2+ Cτ

k∑n=1

‖unδ ‖

2+

k∑n=1

4τ2 ‖Hδ‖2

≤ τ

k∑n=1

‖vnδ ‖2+ C + Cτ,

where we have used the estimates proved before and the bounds on initial data. Nowthe inequality follows from the discrete Gronwall lemma. We proceed with the lastterm in (3.3). To this aim, we need to specify the initial flux Q0

δ = −∇uI + quI ∈(L2(Ω))d. With φ = un

δ − un−1δ , (3.2)1 gives


δ

∥∥2 + τ(∇ ·Qn

δ , unδ − un−1

δ

)+(vnδ − vn−1

δ , unδ − un−1

δ

)= 0.(3.7)

Now take ψ = τQnδ to obtain

τ(Qn

δ −Qn−1δ ,Qn

δ

)− τ

(unδ − un−1

δ ,∇ ·Qnδ

)− τ

(q(un

δ − un−1δ ),Qn

δ

)= 0.

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Further, use (3.7) and rewrite the above left-hand side to obtain

2∥∥un

δ − un−1δ

∥∥2 + τ ‖Qnδ ‖

2 − τ∥∥Qn−1

δ

∥∥2 + τ∥∥Qn

δ −Qn−1δ

∥∥2= 2τ(q(un

δ − un−1δ ),Qn

δ )− 2(vnδ − vn−1δ , un

δ − un−1δ ).

The right-hand side can be estimated using Young’s inequality to get

(3.8)

k∑n=1


δ

∥∥2 + τ∥∥∥Qk

δ

∥∥∥2 + τ

k∑n=1

∥∥Qnδ −Qn−1

δ

∥∥2

≤ Cτ + Cτ2k∑

n=1

‖Qnδ ‖

2 + τ∥∥Q0

δ

∥∥2 .The estimate follows now by the discrete Gronwall lemma. Moreover, from (3.8) wealso get the first and second terms in (3.4). Finally, we take φ = ∇ ·Qn

δ in (3.2)1, useYoung’s inequality, and previous estimates to bound the third term in (3.4).

3.1. Enhanced compactness. As will be seen below, the above estimates arenot sufficient to retrieve the desired limiting equations. To complete the proof ofconvergence, stronger compactness properties are needed. These are obtained bytranslation estimates. To this aim, we define the translation in space

�ξf(·) := f(·)− f(·+ ξ), ξ ∈ R2.

With ξ ∈ R2, we consider Ωξ ⊂ Ω such that Ωξ := {x ∈ Ω|dist(x,Γ) > ξ}. In this way,

the translations �ξf(x) with x ∈ Ω are well defined.For reasons of brevity, the norms and the inner products for the translations

should be understood with respect to Ωξ unless explicitly stated otherwise. Let usfirst consider the translation for un

δ .

Lemma 3.3. It holds that∑N

n=1 τ ‖�ξunδ ‖

2 ≤ C|ξ|.Proof. For (3.2)3 we have after translation in space

(�ξQnδ ,ψ)− (�ξu

nδ ,∇ · ψ)− (�ξ(qu

nδ ),ψ) = 0.

We construct an appropriate test function to obtain the estimate above. Take ηn suchthat {

−Δηn = �ξunδ in Ω

ηn = 0 on Γ

and choose ψ = ∇ηn (note that ψ ∈ H(div; Ω)) to obtain

(�ξQnδ ,∇ηn) + (�ξu

nδ ,�ξu

nδ )− (�ξ(qu

nδ ),∇ηn) = 0.(3.9)

Note that ηn satisfies ‖�ηn‖ = ‖�ξunδ ‖, and therefore ‖ηn‖H2(Ω) ≤ C(Ω) ‖�ξu

nδ ‖ .

This implies that translations of ∇ηn are controlled,

‖∇(�ξηn)‖L2(Ω) ≤ C|ξ| ‖∇ · (∇ηn)‖ ≤ C(Ω)|ξ| ‖�ξu

nδ ‖ .(3.10)

Recalling (3.3), this gives

(3.11) ‖∇(�ξηn)‖ ≤ C|ξ|.

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Thus we have the following estimate

τ

N∑n=1

‖�ξunδ ‖

2= τ

N∑n=1

(qunδ ,∇(�ξη

n)) + τ

N∑n=1

(Qnδ ,∇(�ξη

n))

≤ τN∑

n=1

‖Qnδ ‖ ‖∇(�ξη

n)‖ + τN∑

n=1

‖q‖L∞(Ω) ‖unδ ‖ ‖∇(�ξη

n)‖ .

(3.12)

The conclusion follows by (3.3), the Young inequality and (3.11).The translation estimates for vnδ are bounded by those for un

δ . This is the essenceof the next lemma.

Lemma 3.4. The following estimates hold true:

supk=1,...N

∥∥�ξvkδ

∥∥2 + N∑n=1

∥∥�ξ(vnδ − vn−1

δ )∥∥2 ≤ C ‖�ξvI‖2 + Cτ

N∑n=1

‖�ξunδ ‖

2,(3.13)

N∑n=1

‖�ξvnδ ‖

2 ≤ C|ξ|.(3.14)

Proof. With θ = �ξvnδ in (3.2)2, we get(

�ξvnδ −�ξv

n−1δ ,�ξv

nδ

)= τ (�ξr(u

nδ ),�ξv

nδ )− τ

(�ξHδ(v

n−1δ ),�ξv

nδ

).

The last term in the above rewrites as(�ξHδ(v

n−1δ ),�ξv

nδ

)=(�ξHδ(v

n−1δ ),�ξv

n−1δ

)+(�ξHδ(v

n−1δ ),�ξ(v

nδ − vn−1

δ )).

The monotonicity of Hδ implies that the first term on the right-hand side is positive.Rewriting the left-hand side together with the Cauchy–Schwarz inequality for the firstterm on the right-hand side gives

12

(‖�ξv

nδ ‖

2 −∥∥�ξv

n−1δ

∥∥2 + ∥∥�ξ(vnδ − vn−1

δ )∥∥2)

≤ τLr ‖�ξunδ ‖ ‖�ξv

nδ ‖+ τ

(�ξHδ(v

n−1δ ),�ξ(v

nδ − vn−1

δ ))

≤ 12L

2rτ ‖�ξu

nδ ‖

2+ 1

2τ ‖�ξvnδ ‖

2+ τ2

δ2 ‖�ξvn−1δ ‖2 + 1

4‖�ξ(vnδ − vn−1

δ )‖2.

Summing over n = 1, . . . , k (k ∈ {1, . . . , N}) yields

∥∥�ξvkδ

∥∥2 + 1

2

k∑n=1

∥∥�ξ(vnδ − vn−1

δ )∥∥2

≤ ‖�ξvI‖2 + L2rτ

k∑n=1

‖�ξunδ ‖

2 + τk∑

n=1

‖�ξvnδ ‖

2 +k∑

n=1

2τ2

δ2‖�ξv

n−1δ ‖2.(3.15)

Using Lemma 3.3 and Gronwall’s lemma we obtain

(3.16) supk=1,...,N

∥∥�ξvkδ

∥∥2 ≤ Cτ

N∑n=1

‖�ξunδ ‖

2+ ‖�ξvI‖2 .

The estimate (3.13) follows from above and from (3.15), whereas (3.14) is a directconsequence of Lemma 3.3 and the assumptions on vI .

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3.2. Convergence. For proving the convergence of the time discretizationscheme, we consider the sequence of time-discrete quadruples {(un

δ ,Qnδ , v

nδ , w

nδ ), n =

0, . . . , N} solving Problem Pmvf,nδ , and construct a time-continuous approximation

by linear interpolation. In this sense, for t ∈ (tn−1, tn] (n = 1, . . . , N) we define

(3.17) Zτ (t) := znδ(t− tn−1)

τ+ zn−1

δ

(tn − t)

τ,

where Zτ may refer to any of (U τ ,Qτ , V τ ) and znδ the discrete counterpart. Furtherfor completeness, W τ (t) := Hδ(V

τ (t)). The estimates in Lemma 3.2 can be translateddirectly to (U τ ,Qτ , V τ ,W τ ).

Lemma 3.5. A constant C > 0 exists s.t. for any τ and δ = O(√τ) the following

L2(0, T ;L2(Ω)) estimates hold:

‖U τ‖+ ‖V τ‖+ ‖Qτ‖ ≤ C,(3.18)

‖∂tU τ‖+ ‖∂tV τ‖+ ‖∇ ·Qτ‖ ≤ C.(3.19)

Proof. Equation (3.18) follows easily from (3.3). For instance,

‖U τ‖2 ≤ 2‖unδ ‖2 + 2‖un−1

δ ‖2 ≤ C

and other estimates follow similarly. To estimate ‖∂tV τ‖L2(0,T ;L2(Ω)) we note that,

whenever t ∈ (tn−1, tn], ∂tVτ =

vnδ −vn−1

δ

τ implying

∫ T

0

‖∂tV τ‖2dt =N∑

n=1

∫ tn

tn−1

1

τ2‖vnδ − vn−1

δ ‖2dt ≤N∑

n=1

1

τ‖vnδ − vn−1

δ ‖2 ≤ CτN ≤ C,

where we have used the estimate (3.3). The proof for ∂tUτ is the same as above and

uses the estimate (3.4). The only remaining part in (3.19) is to show that ∇ · Qτ ∈L2(0, T ;L2(Ω)). To see this note that

∇ ·Qτ = ∇ ·Qn−1δ +

t− tn−1

τ∇ · (Qn

δ −Qn−1δ ).

Obtaining the bounds is now a simple exercise involving the a priori estimates alreadyobtained for ∇ ·Qn

δ above in Lemma 3.2. We spare the details.Note that the estimates above are uniform in τ , if δ = O(

√τ ) and we have

(U τ ,Qτ , V τ ,W τ ) ∈ V × S × V × L∞(ΩT ). Moreover, we have the following lemma.Lemma 3.6. A quadruple (u,Q, v, w) ∈ V × S × V × L∞(ΩT ) exists s.t. along a

sequence τ ↘ 0 (and with δ = O(τ12 )) we have by compactness arguments

1. U τ ⇀ u weakly in L2((0, T );L2(Ω)),2. ∂tU

τ ⇀ ∂tu weakly in L2((0, T );L2(Ω)),3. Qτ ⇀ Q weakly in L2((0, T );L2(Ω)d),4. ∇ ·Qτ ⇀ ∇ ·Q weakly in L2((0, T );L2(Ω)),5. V τ ⇀ v weakly in L2((0, T );L2(Ω)),6. ∂tV

τ ⇀ ∂tv weakly in L2((0, T );L2(Ω)),7. W τ ⇀ w weakly-star in L∞(Ω).

In the above only weak convergence of U τ in L2(0, T ;L2(Ω)) is obtained, whichis not sufficient for passing to the limit for nonlinear term r(U τ ). To obtain strongconvergence, we use translation estimates as derived in Lemma 3.3.

Lemma 3.7. It holds that U τ → u strongly in L2((0, T );L2(Ω)).

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Proof. The proof relies on applying the Riesz–Frechet–Kolmogorov theorem (see,e.g., Theorem 4.26 in [8]). In view of ∂tU

τ ∈ L2(0, T ;L2(Ω)), the translation in timeis already controlled. What we need is to control the translation in space:

Iξ :=∫ T

0

∫Ωξ

|�ξUτ |2 dxdt → 0 as |ξ| ↘ 0.

The definition of U τ immediately implies that |Iξ| ≤∑N

n=1τ(2‖�ξunδ ‖2+2‖�ξu

n−1δ ‖2).

Using Lemma 3.3 we find that |Iξ| ≤ C|ξ|, where C is independent of τ and δ, implyingthe strong convergence.

To identify w with H(v) we further need the strong convergence of V τ .Lemma 3.8. For V τ , it holds that V τ → v strongly in L2((0, T );L2(Ω)).Proof. The proof is similar to the above proof of Lemma 3.7 and uses the estimate

in Lemma 3.4. The details are spared.

3.3. The limit equations. Once the strong convergence is obtained, the fol-lowing theorem provides the existence of the weak solution in the mixed variationalformulation.

Theorem 3.9. The limit quadruple (u,Q, v, w) is a solution in the sense ofDefinition 2.1.

Proof. By the weak convergence, the estimates in Lemma 3.5 carry over for thelimit quadruple (u,Q, v, w). Moreover, the time-continuous approximation in (3.17)satisfies

(∂tUτ , φ) + (∇ ·Qτ , φ) + (∂tV

τ , φ) = (∇ · (Qτ −Qnδ ), φ) ,

(3.20)

(∂tVτ , θ)− (r(U τ )−W τ , θ) =

(Hδ(V

τ )−Hδ(vn−1δ ), θ

)+ (r(un

δ )− r(U τ ), θ)(3.21)

(Qτ ,ψ)− (U τ ,∇ ·ψ)− (qU τ ,ψ) = (Qτ −Qnδ ,ψ)− (U τ − un

δ ,∇ ·ψ)(3.22)

− (q(U τ − unδ ),ψ)

for all (φ, θ,ψ) ∈ (L2(0, T ;H10 (Ω)),V ,S). Note that, in fact, (3.20) also holds for

φ ∈ V . Here we choose a better space to identify the limit, where we prove that theterm on the right is vanishing along a sequence τ ↘ 0. By density arguments, thelimit will hold for φ ∈ V .

Consider first (3.20) and note that by Lemma 3.6, the left-hand side convergesto the desired limit. It only remains to show that the right-hand side, denoted byI1, vanishes as τ ↘ 0. Integrating by parts, which is allowed due to the choice ofφ ∈ L2(0, T ;H1

0(Ω)), one has

|I1| ≤(

N∑n=1

τC‖Qnδ −Qn−1

δ ‖2) 1

2(∫ T

0

‖∇φ‖2dt) 1

2

→ 0

due to the estimate (3.4).Next, we consider (3.21). First we prove that the last two integrals on the right-

hand side, denoted by I2 and I3, vanish. For I2 we use the Lipschitz continuity ofHδ and the definition of V τ to obtain

|I2| ≤(

N∑n=1

τ

δ2‖vnδ − vn−1

δ ‖2) 1

2(∫ T

0

‖θ‖2dt) 1

2

.

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Using (3.3) we have |I2| ≤ C τδ (∫ T

0‖θ‖2dt) 1

2 . By the choice of δ, τδ ↘ 0 as τ ↘ 0,

implying that I2 vanishes in the limit. For I3 we use the Lipschitz continuity of rand (3.4) to conclude the same.

For the first term on the left in (3.21), the limit is straightforward. For the limitof the second term, with strong convergence of U τ and weak-* convergence of W τ weget

limτ↘0

(r(U τ )−W τ , θ) = (r(u)− w, θ) ,

leading to the limiting equation

(∂tv, θ) = (r(u)− w, θ) for all θ ∈ V .(3.23)

Now we consider (3.22) and denote the corresponding integrals on the right-hand side,respectively, by I4, I5, and I6. By the definition of Qτ and (3.4), as τ ↘ 0 we obtain

|I4| ≤(

N∑n=1

τ‖Qnδ −Qn−1

δ ‖2) 1

2(∫ T

0

‖ψ‖2dt) 1

2

→ 0.

Similarly, I5 and I6 vanish in the limit using an a priori estimate for unδ − un−1

δ as inLemma 3.2. With this the limit equation takes the form

(Q,ψ)− (u,∇ · ψ)− (qu,ψ) = 0.(3.24)

To conclude the proof what remains is to show that w = H(v). Since we have V τ

strongly converging, we also obtain V τ → v pointwise a.e. and further, as τ ↘ 0,by construction δ ↘ 0. For the set R+ := {(t, x) : v(t, x) > 0}, let us assumeμ := v(t, x, z)/2 > 0. Then the pointwise convergence implies the existence of anεμ > 0 such that V τ > μ for all ε ≤ εμ. Then for any ε ≤ εμ we haveW τ = 1 implyingw = 1. A similar conclusion also holds for R−, where R− := {(t, x) : v(t, x) < 0}.For the case when v = 0; consider the set R0 := {(t, x, z) : v(t, x, z) = 0}. Now in theinterior of the set R0, ∂tv = 0. Next, from the weak convergence of ∂tV

τ ,W τ , r(U τ ),we have the following limit equation

(∂tv, θ) = (r(u) − w, θ) .

Hence, for the interior of the set R0, we obtain w = r(u). Furthermore, the bounds0 ≤ W τ ≤ 1 with weak-∗ convergence of W τ

h to w imply the same bounds on w andhence, w = r(u) with 0 ≤ r(u) ≤ 1.

4. The fully discrete formulation. Following the semidiscrete scheme, wenow consider the fully discrete system (discretized in both space and time) and showits convergence. The steps for the proof of convergence are similar to the semidiscretesituation and whereever the proof is similar, we suppress the details. Further, tosimplify notation, henceforth, we suppress the subscript δ.

Starting with u0h = uI , v

0h = vI , with n ∈ {1, . . . , N}, the approximation

(unh, v

nh ,Q

nh, w

nh) of (u(tn), v(tn),Q(tn), w(tn)) at t = tn solves the following problem.

Problem Pnh. Given (un−1

h , vn−1h ) ∈ (Vh,Vh) find (un

h, vnh ,Q

nh, w

nh) ∈

(Vh,Vh,Sh, L∞(Ω)) satisfying

(unh − un−1

h , φ) + τ(∇ ·Qnh, φ) + (vnh − vn−1

h , φ) = 0,

(vnh − vnh , θ)− τ(r(unh), θ)− τ(Hδ(v

n−1h ), θ) = 0,(4.1)

(Qnh,ψ)− (un

h,∇ · ψ)− (qhunh,ψ) = 0

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for all (φ, θ,ψ) ∈ Vh ×Vh×Sh. For completion, we define wnh = Hδ(v

nh ). For stability

reasons, as before, we choose δ = O(τ12 ) (see [14, 25] for detailed arguments).

From (4.1)1 and (4.1)2, we eliminate vnh , which is computed after having obtained(un

h,Qnh) satisfying for all (φ,ψ) ∈ (Vh,Sh)

(unh − un−1

h , φ) + τ(∇ ·Qnh , φ) + τ(r(un

h)−Hδ(vn−1h ), φ) = 0,

(Qnh,ψ)− (un

h,∇ · ψ)− (qhunh,ψ) = 0.

(4.2)

In the above formulation, the nonlinearities only involve unh; the Hδ is known from

the previous time step and is in L∞.The existence follows from [37, Theorem 4.3], which treats a more general case.

Its proof is based on [41, Lemma 1.4, p. 140]. Following the ideas in section 3, onecan prove that (4.2) has a unique solution pair (un

h,Qnh). This also determines vnh and

wnh uniquely. We summarize the above result.

Lemma 4.1. Problem Pnh has a unique solution quadruple (un

h,Qnh, v

nh , w

nh).

Similar to the semidiscrete case, one can prove the following stability estimates.Lemma 4.2. The following estimates hold:

supk=1,...,N

(∥∥ukh

∥∥+ 1

τ

∥∥vkh − vk−1h

∥∥+ ∥∥vkh∥∥+ ∥∥∥Qkh

∥∥∥) ≤ C,(4.3)

N∑n=1

(1

τ

∥∥unh − un−1

h

∥∥2 + ∥∥Qnh −Qn−1

h

∥∥2(4.4)

+ τ ‖∇ ·Qnh‖

2 + τ∥∥∇ · (Qn

h −Qn−1h )

∥∥2) ≤ C.

We continue with the steps analogous to the semidiscrete situation. As in (3.17)we consider the time-continuous approximation by the piecewise linear interpolationsof the time-discrete solutions. Let Zτ

h denote the interpolated solution with Z standingfor U, V, or Q and W τ

h defined analogously. As before, the estimates in Lemma 4.2carry over for the time-continuous approximation (the proof is omitted).

Lemma 4.3. The time-continuous approximations satisfy the following estimates:

‖∂tU τh‖

2 + ‖∇ ·Qτh‖

2 + ‖U τh‖

2 + ‖V τh ‖2 + ‖∂tV τ

h ‖2 + ‖Qτh‖

2 ≤ C,(4.5)

0 ≤ W τh ≤ 1.(4.6)

Here the norms are taken with respect to L2(0, T ;L2(Ω)). The estimates areuniform in τ and δ and furthermore we have (U τ

h ,Qτh, V

τh ,W τ

h ) ∈ V ×S×V ×L∞(Ω).

Clearly, if τ ↘ 0 with δ = O(τ12 ) implies that both δ, τδ ↘ 0. By compactness

arguments, we get from Lemma 4.3 the following convergence result.Lemma 4.4. Along a sequence τ ↘ 0, it holds that1. U τ

h ⇀ u weakly in L2((0, T );L2(Ω)),2. ∂tU

τh ⇀ ∂tu weakly in L2((0, T );H−1(Ω)),

3. Qτh ⇀ Q weakly in L2((0, T );L2(Ω)d),

4. ∇ ·Qτh ⇀ χ weakly in L2((0, T );L2(Ω)),

5. V τh ⇀ v weakly in L2((0, T );L2(Ω)),

6. ∂tVτh ⇀ ∂tv weakly in L2((0, T );L2(Ω)),

7. W τh ⇀ w weakly-star in L∞(Ω).

As in the semidiscrete case, identification of the above limit χ with ∇ · Q isobtained via smooth test functions. Note that the above lemma only provides weak

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convergence for U τh , V

τh ; in the wake of nonlinearities, the strong convergence is needed.

However, the techniques from the semidiscrete case cannot be applied directly. Thisis because the translation of a function that is piecewise constant on the given meshneed not be piecewise constant on that mesh. We therefore adopt the finite volumeframework in [19] in order to overcome this difficulty.

4.1. Strong convergence. In what follows, we establish the required strongconvergence of U τ

h followed by that of V τh . We provide the notation used below in the

framework of finite volumes. Let E denote the set of edges of the simplices Th. Also,we have that E = Eint ∪ Eext with Eext = E

⋂∂Ω and Eint = E \ Eext. We adopt the

following notation:(4.7)|T | = the area of T ∈ Th, xi = the center of the circumcircle of T,

lij = the edge between Ti and Tj , dij = the distance from xi to lij , σij =|lij |dij

.

In analogy with the spatially continuous case, we define the following discrete innerproduct for any un

h, vnh ∈ Vh:

(unh, v

nh)h :=

∑Ti∈Th

|Ti|unh,iv

nh,i, (un

h, vnh)1,h :=

∑lij∈E

|σij |(unh,i − un

h,j)(vnh,i − vnh,j).

(4.8)

The discrete inner product gives rise to the discrete H10 norm, which is

(4.9) ‖unh‖21,h =

∑lij∈E

|σij |(unh,i − un

h,j)2.

In [19], the following discrete Poincare inequality is proved: ‖unh‖ ≤ C‖un

h‖1,h, withC independent of h or un

h. Based on Lemma 4 in [19], below we show that thetranslations are controlled by the discrete ‖ · ‖1,h norm.

Lemma 4.5. Let Ω be an open bounded set of R2 and let Th be an admissiblemesh. For a given u defined in Ω and extended to u by 0 outside Ω we have

‖�ξu‖2L2(R2) ≤ ‖u‖21,h|ξ|(|ξ|+ Csize(Th)) for all ξ ∈ R2.(4.10)

This shows that for a sequence {unh} having the discrete H1

0 norm uniformlybounded, the L2-norm of the translations �ξu

nh vanishes uniformly with respect to h

as η ↘ 0. This is an essential step in proving the strong L2-convergence for unh. Here

we only need to show that unh has bounded discrete H1

0 norm.Lemma 4.6. For the sequence un

h, the following inequality holds with C indepen-dent of h and n:

‖unh‖1,h ≤ C(‖Qn

h‖+ ‖unh‖).(4.11)

Proof. The approach is inspired from the semidiscrete situation and is adaptedto the present context by defining an appropriate test function. Define

|Ti|fnh (Ti) :=

∑lij

|lij |dij

(unh,i − un

h,j)(4.12)

and note that by the definition of ‖ · ‖21,h ,

(fnh , u

nh) =

∑i

|Ti|fnh (Ti)u

nh(Ti) =

∑lij

|lij |dij

|unh,i − un

h,j|2 = ‖unh‖21,h.(4.13)

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Further, by using the Cauchy–Schwarz we obtain

‖fnh ‖2L2(Ω) =

∑i

|Ti||fnh (Ti)|2 ≤

∑lij

|lij |dij

(uni − un

j )2 1

|Ti|∑lij

|lij |dij

which implies that ‖fnh ‖ ≤ ‖un

h‖1,h. Note that fnh ∈ L2(Ω) and hence, there exists

ψh ∈ Sh which satisfies

∇ · ψh = fnh in Ω,(4.14)

ψh = 0 on Γ.(4.15)

By the bounds on fnh above, it also holds that ‖ψh‖L2(Ω) ≤ C‖fn

h ‖L2(Ω) ≤ C‖unh‖1,h.

By (4.13), (unh,∇·ψh) = (un

h, fnh ) = ‖un

h‖21,h. Now choose for the test function ψ = ψh

in (4.1)3 to obtain

‖unh‖21,h = (un

h,∇ ·ψh) = (Qnh,ψh)− (qhu

nh,ψh)

≤ ‖Qnh‖‖ψh‖+Mq‖un

h‖‖ψh‖ ≤ C‖Qnh‖‖un

h‖1,h + CMq‖unh‖‖un

h‖1,hand the conclusion follows.

In view of the above lemma, obtaining the relative compactness in L2 is straight-forward.

Lemma 4.7. Along a sequence (τ, h) converging to (0, 0) (and with δ = O(√τ)),

U τh converges strongly in L2(0, T ;L2(Ω)).

Proof. Since ∂tUτh is in L2, the translation with respect to time is already con-

trolled. What remains is to consider the translation with respect to space. Take (4.11)and sum over n = 1, . . . , N to obtain

τ

N∑n=1

‖unh‖21,h ≤ Cτ

N∑n=1

(‖Qnh‖2 + ‖un

h‖2) ≤ C(4.16)

using (4.3). Now use Lemma 4.5 to control the translations by the ‖ · ‖1,h norm (afterextending un

h by 0 outside Ω; for simplicity retain the same notation):

τ

N∑n=1

‖unh(·+ ξ)− un

h‖2L2(R2) ≤ C|ξ|(|ξ| + size(Th)),

which, in turn, provides a similar estimate for U τh

τN∑

n=1

‖U τh (·+ ξ)− U τ

h‖2L2(R2) ≤ C|ξ|(|ξ|+ size(Th)).

The Riesz–Frechet–Kolmogorov compactness theorem proves the assertion.The strong convergence of U τ

h leads to the strong convergence of V τh .

Lemma 4.8. Along a sequence (τ, h) converging to (0, 0), V τh converges strongly

to v in L2(0, T ;L2(Ω)).Proof. As before, the translations with respect to time are already controlled by

virtue of ∂tVτh ∈ L2. We now consider the case for the translation with respect to

space. Since both unh, v

nh are piecewise constants in each simplex T , we have for every

x ∈ T

vnh(x) = vn−1h (x) + τ

(r(un

h(x)) − τHδ(vn−1h (x))

),

vnh (x+ ξ) = vn−1h (x+ ξ) + τ

(r(un

h(x+ ξ))− τHδ(vn−1h (x+ ξ))

),

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so that for any x ∈ Ωξ we have �ξ(vnh − vn−1

h ) = τ�ξr(unh) − τ�ξHδ(v

n−1h ). Multi-

plying by �ξvnh and rewriting the left-hand side, we have

(4.17)1

2

{|�ξv

nh |2 − |�ξv

n−1h |2 + |�ξ(v

nh − vn−1

h )|2}

≤ τLr|�ξunh||�ξv

nh | − τ

(�ξHδ(v

n−1h )

)�ξv

nh .

Rewriting the last term to exploit monotonicity of Hδ,

1

2

{|�ξv

nh |2 − |�ξv

n−1h |2 + |�ξ(v

nh − vn−1

h )|2}

≤ τL2r|�ξu

nh|2 +

1

4|�ξv

nh |2 +

1

4|�ξ(v

nh − vn−1

h |2 + τ2

δ2|�ξv

n−1h |2.

Integrating over Ωξ and summing over n = 1, . . . , k for any k ∈ {1, . . . , N} gives

1

2‖�ξv

kh‖2 +

1

4

k∑n=1

‖�ξ(vnh − vn−1

h )‖2

≤ ‖�ξvI,h‖2 + τk∑

n=1

L2r‖�ξu

nh‖2 +

1

4τ

k∑n=1

‖�ξvnh‖2 +

k∑n=1

τ2

δ2‖�ξv

n−1h ‖2,

where the norms are taken with respect to Ωξ. Choosing δ = O(τ12 ) leads to

1

2‖�ξv

kh‖2 +

1

4

k∑n=1

‖�ξ(vnh − vn−1

h )‖2

≤ ‖�ξvI,h‖2 + τ

k∑n=1

L2r‖�ξu

nh‖2 + Cτ

k∑n=1

‖�ξvnh‖2.

Applying Gronwall’s lemma, supk=1,...,N ‖�ξvkh‖2 ≤ C‖�ξvI,h‖2+ τ

∑Nn=1 ‖�ξu

nh‖2.

The strong convergence of U τh in L2(0, T ;L2(Ω)) implies that the last term vanishes

in the limit of |ξ| ↘ 0 (see the proof of Lemma 4.7). To estimate the translations forthe initial condition we consider vI,h as the finite volume approximation of vI defined(formally) by −ΔvI,h = −ΔvI in Ω, with homogenous Dirichlet boundary conditions.This implies ‖v‖1,h ≤ C‖∇vI‖ ≤ C. Since the translations are approaching 0 if‖vI,h‖1,h ≤ C (uniformly in h), ‖�ξvI,h‖ → 0 as |ξ| ↘ 0. From the above weconclude that ‖�ξv

kh‖2 → 0 as |ξ| goes to 0. Finally, note that the definition of V τ

h

implies the rough estimate

∫ T

0

‖�ξVτh ‖2dt ≤ 2τ

N∑n=1

‖�ξvnh‖2 + 2τ

N∑n=1

‖�ξvn−1h ‖2,

and the right-hand side vanishes uniformly in h as |ξ| ↘ 0; hence V τh converges

strongly.

4.2. The limit equations. Up to now we obtained the convergence of thefully discrete triples (U τ

h , Vτh ,Qτ

h) along a sequence (τ, h) approaching (0, 0) withδ = O(

√τ ). Clearly, the (L∞ weakly-star) convergence extends to the sequence

W τh = Hδ(V

τh ). In what follows, we identify the limit discussed in the preceding

section as the weak formulation (2.9).

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Theorem 4.9. The limit quadruple (u,Q, v, w) is a weak solution in the senseof Definition 2.1.

Proof. By the weak convergence, the estimates in Lemma 4.3 carry over for thelimit triple (u,Q, v). By (4.1)1 we have(4.18)∫ T

0

(∂tUτh , φ)dt+

∫ T

0

(∇ ·Qτh, φ)dt+

∫ T

0

(∂tVτh , φ)dt

=

N∑n=1

∫ tn

tn−1

(∂tUτh , φ− φh)dt+

N∑n=1

∫ tn

tn−1

(∇ ·Qτh −∇ ·Qn

h, φ)dt

+N∑

n=1

∫ tn

tn−1

(∇ ·Qτh, φ− φh)dt+

N∑n=1

∫ tn

tn−1

(∇ ·Qτh −∇ ·Qn

h, φh − φ)dt

+

N∑n=1

∫ tn

tn−1

(∂tVτh , φ− φh)dt

for all φ ∈ L2(0, T ;H10 (Ω)), and where φh is the projection φh = Pφ introduced in

section 2.1. Note that we assume again an H1 regularity in space for the test functionφ. We use this to control the terms involving ‖φ − φh‖ by using the property (2.7).A usual density argument lets the result hold for all φ ∈ V .

The left-hand side gives the desired limit terms; it only remains to show that theright-hand side vanishes in the limit. Denote the successive integrals on the right byIi, i = 1, . . . , 5. We deal with each term separately.

For I1 we use (4.5) to obtain that as h ↘ 0

|I1| ≤ ‖∂tU τh‖L2(0,T ;L2(Ω))

(N∑

n=1

∫ tn

tn−1

‖φ− φh‖2L2(Ω)dt

) 12

≤ Ch‖∇φ‖L2(0,T ;L2(Ω)) → 0.

Similarly, by (4.4), for I2 one gets

|I2| ≤(

N∑n=1

τ‖Qnh −Qn−1

h ‖2) 1

2

(N∑

n=1

∫ tn

tn−1

‖∇φ‖2L2(Ω)dt

) 12

≤ Cτ12 .

Clearly I2 vanishes in the limit of τ ↘ 0. The estimates for I3 are analogous anduse the estimate (4.5). The treatment of I4 and I5 is similar and relies on (4.4) andestimate for ∂tV

τh , respectively.

Next we consider (4.1)2, which we rewrite as∫ T

0

(∂tVτh , θ)dt−

∫ T

0

(r(U τh )−W τ

h , θ) dt

=

∫ T

0

(∂tVτh , θ − θh) dt+

N∑n=1

∫ tn

tn−1

(Hδ(V

τh )−Hδ(v

n−1h ), θ

)dt

+

N∑n=1

∫ tn

tn−1

(Hδ(v

n−1h ), θ − θh

)dt+

N∑n=1

∫ tn

tn−1

(r(unh)− r(U τ

h ), θ) dt

+

N∑n=1

∫ tn

tn−1

(r(unh), θh − θ) dt

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for θ ∈ L2(0, T ;H10 (Ω)) and θh is the Ph projection of θ. A better regularity of θ is

again chosen for identifying the limits and controlling the errors due to the projections.We would retrieve the desired limiting equations once we prove that the integrals onthe right-hand side vanish. Let us denote the successive integrals by Ji, i = 1, . . . , 5,respectively. For J1 we get, by using (4.5) and recalling the projection estimate (2.7),

|J1| ≤(∫ T

0

‖∂tV τh ‖2dt

) 12(

N∑n=1

∫ tn

tn−1

‖(θ − θh)‖2dt) 1

2

≤ Ch‖θ‖L2(0,T ;H10 (Ω))

which vanishes in the limit as h ↘ 0. For J2, we use the definition of W τ andLipschitz continuity of Hδ to obtain

|J2| ≤N∑

n=1

τ1

δ‖vnh − vn−1

h ‖‖θ‖ ≤N∑

n=1

τCτ

δ‖θ‖

by using (4.3); and further, using τ/δ ↘ 0 by the construction of δ we obtain J2 → 0.Next, we consider J3:

|J3| ≤ C

(N∑

n=1

∫ tn

tn−1

‖θ − θh‖2dt) 1

2

≤ Ch‖∇θ‖L2(0,T ;L2(Ω)) → 0 as h ↘ 0

because of (2.7). To continue,

|J4| ≤(

N∑n=1

τL2r‖un

h − un−1h ‖2

) 12(

N∑n=1

τ‖θh‖2) 1

2

→ 0,

|J5| ≤ Lr

(N∑

n=1

τ‖unh‖2dt

) 12(

N∑n=1

∫ tn

tn−1

‖(θ − θh)‖2dt) 1

2

≤ Ch‖θ‖L2(0,T ;H10 (Ω)) → 0,

where we use the estimate (4.4) for J4.Let us consider the next equation, that is, (4.1)3. We have, by realigning the

terms,

(4.19)

∫ T

0

(Qτh,ψ)dt−

∫ T

0

(U τh ,∇ ·ψ)dt−

∫ T

0

(qhUτh ,ψ)dt

=

N∑n=1

∫ tn

tn−1

(Qτh −Qn

h ,ψ)dt+

N∑n=1

∫ tn

tn−1

(Qnh,ψ −ψh)dt

+N∑

n=1

∫ tn

tn−1

(unh − U τ

h ,∇ · ψ)dt+N∑

n=1

∫ tn

tn−1

(Unh ,∇ · (ψh −ψ))dt

+

N∑n=1

∫ tn

tn−1

(qh(unh − U τ

h ),ψ)dt+

N∑n=1

∫ tn

tn−1

(qhunh,ψh −ψ)dt

for all ψ ∈ L2(0, T ;H2(Ω)) and where ψh is chosen as the Πh projection of ψ. Asbefore the left-hand side converges to the desired limits. This is obvious except forthe third term where we use the L∞ and strong convergence of qh. Indeed,∫ T

0

(qhUτh ,ψ)dt =

∫ T

0

(qU τh ,ψ)dt+

∫ T

0

((qh − q)U τh ,ψ)dt(4.20)

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and the first term on the right-hand side passes to the desired limit. We show that thesecond term vanishes in the limit. Note that qh−q ∈ (L∞(Ω))2 and hence, (qh−q)U τ

h

has a weak limit. Now choose ψ ∈ L2(0, T ; (C∞c (Ω))2) so that ψ ∈ (L∞(Ω))2. Now

‖(qh − q)U τh‖(L1(Ω))2 ≤ ‖qh − q‖(L2(Ω))2‖U τ

h‖(L2(Ω))

and use the strong convergence of qh in L2 to conclude that the weak limit is indeed 0.Now we show that the right-hand side of (4.19) vanishes in the limit. Let us denote

the integrals by Ki, i = 1, . . . , 6. The successive terms will be treated as before. Webegin with K1:

|K1| ≤ ‖ψ‖L2(0,T ;H1(Ω))

(N∑

n=1

τ‖Qnh −Qn−1

h ‖2) 1

2

≤ Cτ12

using bounds given in (4.4). Thus, K1 goes to 0 in the limit. For K2, recalling thebound (4.3) and the projection estimate (2.7),

|K2| ≤(

N∑n=1

τ‖Qnh‖2) 1

2(

N∑n=1

∫ tn

tn−1

‖ψ −ψh‖L2(Ω)dt

) 12

≤ Ch.

Similarly, we have

|K3| ≤(

N∑n=1

‖unh − un−1

h ‖2) 1

2(∫ T

0

‖∇ · ψ‖2dt) 1

2

≤ Cτ,

|K4| ≤(

N∑n=1

τ‖unh‖2) 1

2(

N∑n=1

∫ tn

tn−1

‖∇ · (ψ −ψh)‖2dt) 1

2

≤ Ch,

|K5| ≤ Mq

(N∑

n=1

τ‖unh − un−1

h ‖2) 1

2

‖ψ‖L2(0,T ;L2(Ω)) ≤ Cτ,

|K6| ≤ Mq

(N∑

n=1

τ‖unh‖2) 1

2(

N∑n=1

τ‖(ψ −ψh)‖2) 1

2

≤ Ch,

all of which vanish in the limit.The identification of w with H(v) is identical to the semidiscrete case. Note

that the limit quadruple (u,Q, v, w) indeed satisfies (2.9), but for test functions hav-ing a better regularity in space, φ ∈ L2(0, T ;H1

0(Ω)), θ ∈ L2(0, T ;H10(Ω)), and

ψ ∈ L2(0, T ;H2(Ω)). In view of the regularity of u, v,Q, density arguments canbe employed to show that the limit equations also hold for φ ∈ L2(0, T ;L2(Ω)), θ ∈L2(0, T ;L2(Ω)),ψ ∈ L2(0, T ;H(div; Ω)), which completes the proof.

5. Numerical computations. We will study the numerical computations intwo parts: the first part deals with the illustration of the physical characteristics ofthe model and the second, the convergence studies. For the former, we will considerthe reactive processes taking place depending upon the appropriate choice of boundaryand initial conditions. For the convergence studies, we will consider a test problemfor which we have constructed an exact solution. We begin with the illustration ofthe physical properties of the model.

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9.93E-01

7.47E-01

5.00E-01

2.54E-01

7.52E-03

1.00E-01

7.50E-02

5.00E-02

2.50E-02

0.00E+00

Fig. 5.1. Concentration profiles for the solute u (top) and the precipitate v (bottom) at differenttimes: t = 0.05, 0.1, 0.2. Note that initially u = 1; v = 0.2 in Ωv and 0 elsewhere. As t increases,the dissolution front moves rightwards.

5.1. Illustrative computations. Let us take Ω := (0, 1)× (0, 1) and we makethe following choices:

T = 1, q = (0.5, 0), D = 1, r(u) = 2u+(u − 0.5)+; h = 0.0125, τ = 0.005.

For the boundary condition, −ν ·∇u = 0 on Γ\{x = 0}, with the boundary conditionat x = 0 specified in the different cases considered below. Note that for this choice ofprecipitation rate, r(u) = 0 for u ≤ 0.5 and r(1) = 1 so that u = 1 is the equilibriumsolution, that is, for which the dissolution rates balance the precipitation rate. Tospecify the initial condition, let us consider Ωv := (0.2, 0.8) × (0.2, 0.8) with clearlyΩv(⊂ Ω) being a smaller square inside the original domain. We study the followingcases:

Case (a). Equilibrium situation. For the initial conditions, we choose uI =1, vI = 0.1χΩv with χΩv denoting the characteristic function for the set Ωv. Weimpose u = 1 at x = 0 as the boundary condition. With u = 1 being the equilibriumsolution, the initial and the boundary conditions ensure that no changes take place inthe solution as the initial conditions satisfy the equations (2.1)–(2.2). This is easilyconfirmed numerically by computing

‖uI − unh‖ = 1.04e− 11, ‖vI − vnh‖ = 7.98e− 7

in the L2(0, T ;L2(Ω)) norm.Case (b). Dissolution fronts. To initiate the dissolution front, we choose the

boundary condition at x = 0; hence we choose

uI = 1, vI = 0.1χΩv , u = 0 at x = 0.

Clearly, at the boundary one has r(u) −H(v) < 0, and this is propagated inside theentire domain, initiating a dissolution process. The numerical results in Figure 5.1show a depletion of the precipitate and the occurrence of dissolution, with the supportof v shrinking as time proceeds.

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5.2. Convergence studies. We consider a test problem similar to (2.1)–(2.2),but including a right-hand side in the first equation (see [26] where we first announcedpart of these results). This is chosen in such a way that the problem has an exactsolution, which is used then to test the convergence of the mixed finite element scheme.Specifically, for T = 1 and Ω = (0, 5) × (0, 1), and with r(u) = [u]2+ (where [u]+ :=max{0, u}), we consider the problem⎧⎨

⎩∂t(u+ v) +∇ · (qu−∇u) = f in ΩT ,

∂tv = (r(u) − w) on ΩT ,w ∈ H(v) on ΩT .

Here q = (1, 0) is a constant velocity, whereas

f(t, x, y) =1

2ex−t−5

(1− ex−t−5

)− 32

(1− 1

2ex−t−5

)−{

0 if x < t,ex−t−5 if x ≥ t,

and the boundary and initial conditions are such that

u(t, x, y) =(1− ex−t−5

) 12 and v(t, x, y) =

{0 if x < t,ex−t−1

e5 if x ≥ t,

providing w(t, x, y) =

{1 if x < t,1− ex−t−5 if x ≥ t

form a solution triple.We consider the mixed finite element discretization of the problem above, based

on the time stepping in section 4 and the lowest order Raviart–Thomas elementsRT0. The numerical scheme was implemented in the software package ug [6]. Thesimulations are carried out for a constant mesh diameter h and time step τ , satisfyingτ = h. We start with h = 0.2, and refine the mesh (and, correspondingly, τ and δ)four times successively by halving h up to h = 0.0125. We compute the errors for uand v in the L2 norms,

Ehu = ‖u− U τ‖L2(ΩT ), respectively, Eh

v = ‖v − V τ‖L2(ΩT ).

These are presented in Tables 5.1 and 5.2 for two different choices of δ. Althoughtheoretically no error estimates could be given due to the particular character of thedissolution rate, the tabulated results also include an estimate of the convergenceorder, based on the reduction factor between two successive calculations:

α = log2(Ehu/E

h2u ) and β = log2(E

hv /E

h2v ).

Table 5.1

Convergence results for the mixed scheme, with the explicit discretization for v; h = τ andδ =

√τ .

h ‖u− Uτ‖2 α ‖v − V τ‖2 β

0.2 1.1700e-01 1.8409e-010.1 6.414e-02 0.43 9.927e-02 0.450.05 3.396e-02 0.46 5.317e-02 0.450.025 1.726e-02 0.49 2.785e-02 0.470.0125 8.42e-03 0.52 1.420e-02 0.49

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Table 5.2

Convergence results for the mixed scheme, with the explicit discretization for v; h = τ and δ = 5τ .

h ‖u− Uτ‖2 α ‖v − V τ‖2 β

0.2 2.774958e-01 4.300839e-010.1 1.263555e-01 0.57 1.881785e-01 0.590.05 2.353196e-02 1.21 8.905975e-02 0.540.025 1.072494e-02 0.57 1.792503e-02 1.160.0125 2.308967e-03 1.11 4.368191e-03 1.02

Table 5.3

Convergence results for the fully implicit mixed discretization; here h = τ and δ =√τ .

h ‖u− Uτ‖2 α ‖v − V τ‖2 β

0.2 1.031e-01 1.593e-010.1 5.925e-02 0.40 9.023e-02 0.410.05 3.247e-02 0.43 5.031e-02 0.420.025 1.686e-02 0.47 2.703e-02 0.450.0125 8.313e-03 0.51 1.3980e-02 0.47

Table 5.4

Convergence results for the fully implicit mixed discretization; here h = τ and δ = 5τ .

h ‖u− Uτ‖2 α ‖v − V τ‖2 β

0.2 2.522029e-01 3.905949e-010.1 1.172746e-01 0.55 1.724693e-01 0.590.05 3.964776e-02 0.78 6.064739e-02 0.750.025 1.047443e-02 0.96 1.739749e-02 0.900.0125 2.282752e-03 1.10 4.302130e-03 1.01

The tests are carried out with two choices of δ: δ =√τ (which is supported by the

theory) and δ = 5τ (still providing stability, but without having any rigorous conver-gence proof). As resulting from Tables 5.1 and 5.2, for this test case it appears thatthe method converges sublinearly, respectively, linearly. This suggests that practicallyδ = O(τ) (by maintaining, however, the stability of the scheme) is better.

Similar results are observed for an implicit discretization for v (although thisscheme is not analyzed here; this can be obtained with minor modifications of theproofs here). The implicit scheme provides a set of coupled nonlinear equation forthe triple (un

h,Qnh, v

nh). A Newton iteration is used to solve the resulting system (see

[36, 38], where the Newton method is applied to similar problems). The tests areapplied to the case described before, and the results are presented in Tables 5.3 and5.4. As in the semiimplicit scheme, we see that for the test problem the convergencerate is sublinear for the case δ =

√τ and linear for δ = 5τ .

6. Conclusions. We have considered the semidiscrete and fully discrete numer-ical methods for the upscaled equations. These equations describe the transport andreactions of the solutes. The numerical methods are based on a mixed variational for-mulation where we have a separate equation for the flux. These numerical methodsretain the local mass conservation property. The reaction terms are nonlinear andthe dissolution term is multivalued described by a Heaviside graph. To avoid dealingwith the inclusions, we use the regularized Heaviside function with the regularizationparameter δ dependent on the time step τ . This implies that in the limit of vanishingdiscretization parameters automatically yields δ ↘ 0. For the fully discrete situation,we have used lowest order Raviart–Thomas elements. The convergence analysis of

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both formulations have been proved using compactness arguments, based on trans-lation estimates. In particular, a discrete H1

0 norm is used in the proof for the fullydiscrete scheme.

The work is complemented by the numerical experiments where we study someillustrative examples exhibiting the physical properties of the model. Further, a testcase is considered where we construct an exact solution and compare the numericalsolution. This numerical study provides us convergence rates for the problem underconsideration.

Appendix A. Existence of solution for Pmvf,nδ . In this appendix, we prove

the existence of a solution for ProblemPmvf,nδ . We keep τ and δ fixed and let h ↘ 0 in

the fully discrete problem Pnh. The limit will solve Problem Pmvf,n

δ . All the steps aresimilar to the fully discrete case discussed before; therefore, we only give the outlineof the proof. Along a sequence h ↘ 0, Lemma 4.2 provides the following convergenceresults:

1. unh ⇀ un

δ weakly in L2(Ω),2. Qn

h ⇀ Qnδ weakly in L2(Ω)d,

3. ∇ ·Qnh ⇀ χ weakly in L2(Ω),

4. vnh ⇀ vnδ weakly in L2(Ω).As before, identification of χ with ∇·Qn

δ takes place via standard arguments. Further,Lemma 4.6 with the estimate (4.3) gives ‖un

h‖1,h ≤ C and after extending unh by 0,

Lemma 4.5 implies

‖�ξunh‖L2(R2) ≤ C|ξ|(|ξ| + size(Th)).

Since the right-hand side vanishes uniformly as |ξ| ↘ 0, the use of the Riesz–Frechet–Kolmogorov compactness theorem yields strong convergence of un

h to unδ . Now one

can use the projection properties and pass h ↘ 0 to show that the limit solves Pmvf,nδ .

Note that having v discretized explicitly in (4.1)2, no nonlinearities in vnh are involvedand therefore there is no need for strong convergence for vnh .

Acknowledgment. Part of the work was completed when K. Kumar visited theInstitute of Mathematics, University of Bergen.

REFERENCES

[1] G. Allaire, R. Brizzi, and A. Mikelic, Two-scale expansion with drift approach to theTaylor dispersion for reactive transport through porous media, Chem. Eng. Sci., 65 (2010),pp. 2292–2300.

[2] T. Arbogast, M. Obeyesekere, and M. F. Wheeler, Numerical methods for the simulationof flow in root-soil systems, SIAM J. Numer. Anal., 30 (1993), pp. 1677–1702.

[3] T. Arbogast and M. F. Wheeler, A characteristics-mixed finite element method foradvection-dominated transport problems, SIAM J. Numer. Anal., 32 (1995), pp. 404–424.

[4] J. W. Barrett and P. Knabner, Finite element approximation of the transport of reactivesolutes in porous media. Part I: Error estimates for nonequilibrium adsorption processes,SIAM J. Numer. Anal., 34 (1997), pp. 201–227.

[5] 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.

[6] P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Reichert, and

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

[7] N. Bouillard, R. Eymard, R. Herbin, and P. Montarnal, Diffusion with dissolution andprecipitation in a porous medium: Mathematical analysis and numerical approximation ofa simplified model, M2AN Math. Model. Numer. Anal., 41 (2007), pp. 975–1000.

Dow

nloa

ded

10/0

8/13

to 1

31.1

55.1

51.8

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



[8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer,New York, 2011.

[9] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Com-putational Mathematics 15, Springer-Verlag, New York, 1991.

[10] E. Cariaga, F. Concha, I. S. Pop, and M. Sepulveda, Convergence analysis of a vertex-centered finite volume scheme for a copper heap leaching model, Math. Methods Appl. Sci.,33 (2010), pp. 1059–1077.

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

[12] C. Dawson and V. Aizinger, Upwind-mixed methods for transport equations, Comput. Geosci.,3 (1999), pp. 93–110.

[13] C. N. Dawson, C. J. van Duijn, and M. F. Wheeler, Characteristic-Galerkin methods forcontaminant transport with nonequilibrium adsorption kinetics, SIAM J. Numer. Anal., 31(1994), pp. 982–999.

[14] V. M. Devigne, I. S. Pop, C. J. van Duijn, and T. Clopeau, A numerical scheme for thepore-scale simulation of crystal dissolution and precipitation in porous media, SIAM J.Numer. Anal., 46 (2008), pp. 895–919.

[15] C. J. van Duijn and P. Knabner, Solute transport through porous media with slow adsorption,in Free Boundary Problems: Theory and Applications, Vol. I (Irsee, 1987), Pitman Res.Notes Math. Ser. 185, Longman Sci. Tech., Harlow, 1990, pp. 375–388.

[16] 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.

[17] C. J. van Duijn and P. Knabner, Travelling wave behaviour of crystal dissolution in porousmedia flow, European J. Appl. Math., 8 (1997), pp. 49–72.

[18] C. J. van Duijn and I. S. Pop, Crystal dissolution and precipitation in porous media: Porescale analysis, J. Reine Angew. Math., 577 (2004), pp. 171–211.

[19] R. Eymard, T. Gallouet, and R. Herbin, Convergence of finite volume schemes for semi-linear convection diffusion equations, Numer. Math., 82 (1999), pp. 91–116.

[20] R. Eymard, D. Hilhorst, and M. Vohralık, A combined finite volume–noncon-forming/mixed-hybrid finite element scheme for degenerate parabolic problems, Numer.Math., 105 (2006), pp. 73–131.

[21] D. Hilhorst and M. Vohralık, A posteriori error estimates for combined finite volume–finiteelement discretizations of reactive transport equations on nonmatching grids, Comput.Methods Appl. Mech. Engrg., 200 (2011), pp. 597–613.

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

[23] P. Knabner, C. J. van Duijn, and S. Hengst, An analysis of crystal dissolution fronts inflows through porous media. Part 1: Compatible boundary conditions, Adv. Water Resour.,18 (1995), pp. 171–185.

[24] K. Kumar, T. L. van Noorden, and I. S. Pop, Effective dispersion equations for reactive flowsinvolving free boundaries at the microscale, Multiscale Model. Simul., 9 (2011), pp. 29–58.

[25] K. Kumar, I. S. Pop, and F. A. Radu, Convergence Analysis for a Conformal Discretiza-tion of a Model for Precipitation and Dissolution in Porous Media, CASA Report 12-08,Eindhoven University of Technology, Eindhoven, The Netherlands, 2012.

[26] K. Kumar, I.S. Pop, and F.A. Radu, Numerical analysis for an upscaled model for dissolutionand precipitation in porous media, in Numerical Mathematics and Advanced Applications,A. Cangiani, R. L. Davidchack, E. Georgoulis, A. N. Gorban, J. Levesley, and M. V.Tretyakov, eds., 2011, Springer Berlin Heidelberg, 2013, pp. 703–711.

[27] I. Metzmacher, F. A. Radu, M. Bause, P. Knabner, and W. Friess, A model describingthe effect of enzymatic degradation on drug release from collagen minirods, European J.Pharmaceutics Biopharmaceutics, 67 (2007), pp. 349–360.

[28] A. Mikelic and C. J. van Duijn, Rigorous derivation of a hyperbolic model for Taylor dis-persion, Math. Models Methods Appl. Sci., 21 (2011), pp. 1095–1120.

[29] P. Moszkowicz, J. Pousin, and F. Sanchez, Diffusion and dissolution in a reactive porousmedium: Mathematical modelling and numerical simulations, J. Comp. Appl. Math., 66(1996), pp. 377–389.

[30] T. L. van Noorden, Crystal precipitation and dissolution in a thin strip, European J. Appl.Math., 20 (2009), pp. 69–91.

[31] T. L. van Noorden, I. S. Pop, and M. Roger, Crystal dissolution and precipitation inporous media: L1-contraction and uniqueness, Discrete Contin. Dynam. Systems, 2007,pp. 1013–1020.

Dow

nloa

ded

10/0

8/13

to 1

31.1

55.1

51.8

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



[32] M. Ohlberger and C. Rohde, Adaptive finite volume approximations for weakly coupledconvection dominated parabolic systems, IMA J. Numer. Anal., 22 (2002), pp. 253–280.

[33] A. Pawell and K.-D. Krannich, Dissolution effects in transport in porous media, SIAM J.Appl. Math., 56 (1996), pp. 89–118.

[34] J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems, Non-linear Anal., 39 (2000), pp. 261–279.

[35] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations,Springer Series in Computational Mathematics 23, Springer-Verlag, Berlin, 1994.

[36] F. A. Radu and I. S. Pop, Newton method for reactive solute transport with equilibriumsorption in porous media, J. Comput. Appl. Math., 234 (2010), pp. 2118–2127.

[37] F. A. Radu, I. S. Pop, and S. Attinger, Analysis of an Euler implicit-mixed finite elementscheme for reactive solute transport in porous media, Numer. Methods Partial DifferentialEquations, 26 (2010), pp. 320–344.

[38] F. A. Radu and I. S. Pop, Mixed finite element discretization and Newton iteration for areactive contaminant transport model with nonequilibrium sorption: Convergence analysisand error estimates, Comput. Geosci., 15 (2011), pp. 431–450.

[39] B. Riviere and M. F. Wheeler, Non conforming methods for transport with nonlinear reac-tion, in Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treat-ment, Contemp. Math. 295, AMS, Providence, RI, 2002, pp. 421–432.

[40] J. Rubin, Transport of reacting solutes in porous media: Relation between mathematical natureof problem formulation and chemical nature of reaction, Water Res. Resour., 19 (1983),pp. 1231–1252.

[41] R. Temam, Navier-Stokes equations: Theory and Numerical Analysis, Stud. Math. Appl. 2,3rd ed., North-Holland, Amsterdam, 1984.

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://w

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.sia

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Eindhoven University of Technology · pore-scale simulation of crystal dissolution and precipitation in porous media in Free Boundary Problems: Theory and Applications, Vol. I (Irsee,