A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on nonmatching grids

A posteriori error estimates for combined finite volume–finite element discretizations of reactive transport equations on

nonmatching grids

Danielle Hilhorst¹, and Martin Vohral´ık²

1 Laboratoire de Math´ematiques, Analyse Num´erique et EDP, Universit´e de Paris-Sud et CNRS, Bˆat. 425, 91 405 Orsay, France. danielle.hilhorst@math.u-psud.fr

2 UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France &

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France.

vohralik@ann.jussieu.fr

Abstract

We derive fully computable a posteriori error estimates for vertex-centered finite volume-type discretizations of transient convection–diffusion–reaction equations. Our estimates enable actual control of the error measured either in the energy norm or in the energy norm augmented by a dual norm of the skew-symmetric part of the differential operator. Lower bounds, global-in-space but local-in-time, are also derived. These lower bounds are fully robust with respect to convection or reaction dominance and the final simulation time in the augmented norm setting. On the basis of the derived estimates, we propose an adaptive algorithm which enables to automatically achieve a user-given relative precision. Moreover, this algorithm leads to optimal efficiency as it balances the time and space error contributions. As an example, we apply our estimates to the combined finite volume–finite element scheme, including such features as use of mass lumping for the time evolution or reaction terms, of upwind weighting for the convection term, and discretization on nonmatching meshes possibly containing nonconvex and non-star-shaped elements. Numerical experiments illustrate the theoretical developments.

Key words: transient convection–diffusion–reaction equation, finite volume method, finite ele- ment method, convection dominance, inhomogeneous and anisotropic diffusion, nonmatching grids, a posteriori error estimates, guaranteed upper bound, adaptive mesh refinement

AMS subject classifications: 65M15, 65M60, 76M12, 76S05

∗This work was supported by the GNR MoMaS (PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN), France.

1 Introduction

We consider in this paper the time-dependent convection–diffusion–reaction problem

u_t− ∇ ·(S∇u) +∇ ·(uv) +ru=f in Ω×(0, T), (1.1a)

u(·,0) =u₀ in Ω, (1.1b)

u= 0 on ∂Ω×(0, T), (1.1c)

where S is in general an inhomogeneous and anisotropic (nonconstant full-matrix) diffusion–

dispersion tensor, v is a (dominating) velocity field, r is a reaction function, f is a source term, u₀ prescribes the initial condition, Ω⊂R^d,d= 2,3, is a polygonal (polyhedral for d= 3) domain (open, bounded, and connected set), and (0, T) is a time interval. Our purpose is to derive a pos- teriori error estimates for conforming locally conservative discretizations of problem (1.1a)–(1.1c).

A particular attention will be payed to the combined finite volume–finite element scheme [13] on nonmatching grids.

One of the first works on a posteriori error estimates for finite element discretizations of steady convection–diffusion–reaction problems are those of Angermann [2] and of Eriksson and John- son [9]. In these works, the overestimation factor depends unfavorably on the ratio between convection and diffusion. Estimates with semi-robust lower bounds in the energy norm and esti- mates with robust lower bounds in the energy norm augmented by the dual norm of the material derivative were then derived by Verf¨urth respectively in [23] and [26]. The robustness result has been extended to the unsteady case in [25]. Recently, attention has also been paid to vertex- centered finite volume methods. Let us mention, in the steady convection–diffusion–reaction case and energy norm setting, Lazarov and Tomov [17], Carstensen et al. [7], Nicaise [18], [30], and Ju et al. [16]. Fewer results are known in the unsteady case. L¹-norm estimates for nonlinear problems are derived by Ohlberger [19], whereas the energy norm setting has been pursued in, e.g., Felcman and Kubera [15] or Amaziane et al. [1]. Typically, the estimate only gives the error upper bound up to an undetermined constant, so that the actual error control is not possible.

The present paper aims at enriching the known results for vertex-centered finite volume-type methods in several directions. Firstly, we derive fully and easily computable estimates, not fea- turing any undetermined constants. We achieve this by introducing H(div,Ω)-conforming locally conservative diffusive and convective flux reconstructions, following [29,31,10,11]. Consequently, the estimates allow for actual error control in unsteady convection–diffusion–reaction problems.

Secondly, we propose a space-time adaptive algorithm which equilibrates properly the time and space contributions in the sense that they are in actual balance, not weighted by any undeter- mined quantities. Moreover, this algorithm is proposed to guarantee a user-given relative precision in the simulation. Thirdly, following [25], we also prove a lower error bound. Fourthly, our results are valid for a larger family of conforming locally conservative discretizations, in the framework of the so-called combined finite volume–finite element method, cf. Feistauer et al. [14] and the references therein. Consequently, the analysis includes such features as use of mass lumping for the time evolution or reaction terms. Lastly, we also treat the case of very general nonmatch- ing meshes possibly containing nonconvex and non-star-shaped elements; convergence analysis of the combined finite volume–finite element method on such meshes (for degenerate parabolic convection–diffusion–reaction problems) was carried out in [13].

The paper is organized as follows: in Section 2, the continuous problem is described. Sec- tion3collects the notation of the discrete setting and introduces the combined finite volume–finite element scheme. In Sections 4 and 5, the a posteriori error estimate and its efficiency is respec- tively stated and proved. Finally, an adaptive algorithm is presented in Section 6 and numerical experiments in Section 7.

2 The continuous setting

We state in this section our assumptions on the data and define a weak solution of problem (1.1a)–

(1.1c).

2.1 Assumptions on the data

Let the time interval (0, T] be split such that 0 = t₀ < . . . < tn < . . . < tN = T and define τn := tn −t_n−1, n ∈ {1, . . . , N}. On each time interval (t_n−1, tn], n ∈ {1, . . . , N}, consider a partition Dhⁿ of Ω into closed polygons such that Ω =S

D∈D_hⁿDand such that the intersection of the interiors of two different polygons is empty. Denote the set of its sides (edges for d= 2 and faces for d= 3) byFhⁿ. Also set QT := Ω×(0, T). In the sequel, we will use the notation |R|for the d^′-dimensional Lebesgue measure of R⊂R^d′; at the same time |R|stands for the cardinality of a set R.

We suppose that the data of problem (1.1a)–(1.1c) satisfy:

Assumption (A) (Data)

(A1) S_ij ∈L^∞(Q_T), 1≤i, j≤d, is a symmetric, bounded, and uniformly positive definite tensor such that for all n∈ {1, . . . , N} and for all D∈ Dⁿh,

CSⁿ,Du·u≥S(x, t)u·u≥cⁿS,Du·u, CSⁿ,D >0, cⁿS,D >0 ∀u∈R^d for a.e. x∈D and a.e. t∈(t_n−1, tn];

(A2) v∈L^∞(QT) such that ∇ ·v∈L²(0, T;L^∞(Ω)) such that for all n∈ {1, . . . , N} and for all D∈ Dⁿh,

|v(x, t)| ≤Cvⁿ,D, Cvⁿ,D≥0 for a.e. x∈D and a.e. t∈(tn−1, tn];

(A3) r∈L^∞(QT);

(A4) For all n∈ {1, . . . , N} and for all D∈ Dhⁿ,

1

2∇ ·v(x, t) +r(x, t)≥cⁿv,r,D,∇ ·v(x, t) +r(x, t)≤Cvⁿ,r,D, r(x, t)≤C_r,Dⁿ , cⁿv,r,D ≥0, Cvⁿ,r,D≥0, C_r,Dⁿ ≥0

for a.e. x∈D and a.e. t∈(t_n−1, tn];

(A5) f ∈L²(QT);

(A6) u₀∈L^∞(Ω);

(A7) if cⁿ_v,r,D = 0, then C_vⁿ_,r,D = 0.

2.2 Continuous problem

Let X := L²(0, T;H₀¹(Ω)), X^′ =L²(0, T;H⁻¹(Ω)), and Z := {v ∈ X;v_t ∈ X^′}. Let v ∈ Z. We define B(v)∈X^′ by

hB(v), ϕi:=hvt, ϕi+ (S∇v,∇ϕ) + (∇ ·(vv), ϕ) + (rv, ϕ) (2.1) for all ϕ∈ H₀¹(Ω) and a.e. t∈ (0, T). Hereh·,·i stands for the duality pairing between H⁻¹(Ω) and H₀¹(Ω) and (·,·) for the L²(Ω) scalar product. We will also use (·,·)R for the L²(R) scalar

product on R ⊂Ω and h·,·i^R for the L²(R) scalar product onR ⊂R^d−1. We say that a function u is a weak solution of problem (1.1a)–(1.1c) if u∈Z,u(·,0) =u₀ inH⁻¹(Ω), and u satisfies the equality

hB(u), ϕi= (f, ϕ) (2.2)

for a.e. t∈(0, T) and for allϕ∈H₀¹(Ω).

Forv∈X, we introduce the space-time energy norm kvk²X :=

Z T

0 |||v(·, t)|||²dt,

|||v|||² :=S¹²∇v²+(¹₂∇ ·v+r)¹²v²,

(2.3)

where k · kdenotes the L²(Ω) norm.

Assumption (A) then implies

1

2kv(·, T)k²+kvk²X ≤ Z T

0 hB(v), vidt+¹₂kv(·,0)k² ∀v∈Z. (2.4) Under Assumptions (A), there in particular exists a unique solution of (2.2).

Define, forv∈Z,

b_A(v) :=∇ ·(vv)−¹₂(∇ ·v)v, (2.5) and, for all ϕ∈H₀¹(Ω) and a.e. t∈(0, T), BS(v),BA(v)∈X^′ by

hBS(v), ϕi:= (S∇v,∇ϕ) + ((¹₂∇ ·v+r)v, ϕ),

hBA(v), ϕi:= (b_A(v), ϕ). (2.6)

We will in the sequel also need a spaceY, inspired by that of [25]: Y :={v∈X;∂tv+bA(v)∈X^′} that we equip with its graph norm

kvk²Y :=kvk²X +k∂tv+bA(v)k²X^′. (2.7) Recall that

k∂tv+bA(v)k^X′ = sup

ϕ∈X;kϕkX=1

Z T

0 h∂tv+bA(v), ϕi(t) dt.

We will also localize the k · k^X and k · k^Y norms on (tn−1, tn],n∈ {1, . . . , N}, as follows:

kvk²X(tn−1,tn) :=

Z tn

tn−1

|||v(·, t)|||²dt,

kvk²Y(tn−1,tn) := kvk²X(tn−1,tn)+k∂tv+bA(v)k²X^′(tn−1,tn).

3 The combined finite volume–finite element method

We define in this section admissible grids and the combined finite volume–finite element scheme.

This section can be skipped if the reader is only interested in a posteriori error estimates for conforming locally conservative methods and not in the details of the present scheme.

•

•

•

•

•

•

• •

Ω Thⁿ

Dⁿh

V_E

V_D

E

D V_F

σ_D,E

σE,F

Figure 1: Primal nonmatching grid Dhⁿ (dashed) and dual triangular gridThⁿ (solid) with D, E ∈ Dhⁿ,VD, VE ∈ Vh^n,int,VF ∈ Vh^n,ext,σD,E =∂D∩∂E∈ Fh^n,int, and σE,F ∈ Fh^n,ext

3.1 Grids

We call the grid Dhⁿ introduced in Section 2.1 a primal grid of Ω. In particular we admit non- matching grids, i.e., the case where there exist two different polygons D, E ∈ Dⁿh such that their intersection is not an empty set but it is not a common vertex, edge, or face of D and E. We also allow for nonconvex elements and elements which are not star-shaped. An example of an admissible primal grid is given in Figure 1 by the dashed line. We suppose that there exists a family of points Vh^n,int such that there is one point VD in the interior of D associated with each D∈ Dhⁿ. ForD∈ Dhⁿ, we denote by F^D the set of subsetsσ of ∂D such that there existsE ∈ Dⁿh

such that σ=σD,E:=∂D∩∂E has a positive (d−1)-dimensional Lebesgue measure. If there is a part of ∂D with a positive (d−1)-dimensional Lebesgue measure lying on the boundary, thenF^D contains in addition a union of σ =σD,E ⊂∂D∩∂Ω covering ∂D∩∂Ω and such that eachσD,E

has a positive (d−1)-dimensional Lebesgue measure and contains exactly one point VE ∈ Vh^n,ext

defined below. We remark that ∂D=P

σ_D,E∈FDσD,E, thatσD,E is not necessarily a (whole) side of D, and that σD,E not necessarily lies in a hyperplane of R^d, see Figure1. We denote byFh^n,int

the union of all σD,E=∂D∩∂E for some D, E∈ Dhⁿ, and by Fh^n,ext the union of all σ⊂∂Ω.

A dual grid of Ω is a partition Thⁿ of Ω into closed simplices which satisfies the following properties: (i) The set of pointsVh^n,int is contained in the set of vertices ofThⁿ, denoted byVhⁿ; (ii) The vertices from V_h^n,ext := Vhⁿ\ V_h^n,int lie on the boundary of Ω; (iii) Thⁿ is conforming, i.e. the intersection of two different simplices is either an empty set or their common vertex, edge, or face;

(iv) Ω = S

K∈T_hⁿK. This definition is not unique: we have a choice in connecting the different pointsVD ∈ Vh^n,int and also a choice in the definition of the vertices on the boundary. The general intention is to find a triangulation such that the transmissibilities Sⁿ

D,E defined below by (3.4) were non-negative, since this implies the discrete maximum principle, see Remark 3.2below. We do not impose any particular simultaneous requirement on Dhⁿ and Thⁿ except of the assumption that the intersection of the sides of Dⁿh and Thⁿ has a zero (d−2)-dimensional Lebesgue measure (in practice, the sides of Dⁿh andThⁿ can intersect but not coincide). An example of a dual grid to a primal nonmatching grid is given in Figure 1 by the solid line. For K ∈ Thⁿ, we denote by hK

the diameter ofK. For a vertexVD ∈ Vh^n,int, we denote byM(VD) the set of all vertices VE ∈ Vhⁿ

such that there exists an edge of the dual grid Thⁿ betweenVD and VE.

3.2 The combined scheme

The combined finite volume–finite element scheme is obtained by the discretization of the diffusion term of (1.1a) by means of the piecewise linear conforming finite element method on Thⁿ, the discretization of the other terms of (1.1a) by means of the cell-centered finite volume method on Dhⁿ, and using an implicit finite difference time stepping.

For the ease of notation, let us denote by VE the boundary vertices, even if there are no corresponding E ∈ Dhⁿ, and let us set uⁿ_E = 0 for all n ∈ {1, . . . , N} and all VE ∈ Vh^n,ext (this corresponds to the discretization of the homogeneous Dirichlet boundary condition). The scheme then reads: find the valuesuⁿ_D,n∈ {1, . . . , N},D∈ Dⁿh, such that

uⁿ_D−uⁿ⁻¹_D

τn |D| − X

V_E∈M(VD)

Sⁿ_D,E(uⁿ_E−uⁿ_D) + X

σ_D,E∈FD

hvⁿ·n,1i^σD,E uⁿ_D,E+r_Dⁿuⁿ_D|D|=f_Dⁿ|D|

∀n∈ {1, . . . , N},∀D∈ Dⁿh.

(3.1)

The value u⁰_D forD∈ Dh¹ is given by u⁰_D := (u₀,1)D/|D|and it corresponds to the discretization of the initial condition. When Dⁿ⁻¹h and Dⁿh (and Thⁿ⁻¹ and Thⁿ) coincide, uⁿ⁻¹_D = uⁿ⁻¹_D , the approximate value from the previous time step. In the opposite case, we define uⁿ⁻¹_D by the value of uⁿ⁻¹_h (cf. its definition (3.8) below) at the pointVD corresponding toD∈ Dⁿh. Let

fⁿ(x) := 1 τ_n

Z tn

tn−1

f(x, t) dt x∈Ω, n∈ {1, . . . , N}, rⁿ(x) := 1

τn

Z tn

t_n−1

r(x, t) dt x∈Ω, n∈ {1, . . . , N}, vⁿ(x) := 1

τn

Z tn

tn−1

v(x, t) dt x∈Ω, n∈ {1, . . . , N},

where n is the unit normal vector of∂D, outward toD. In the above definition, we have also set f_Dⁿ := (fⁿ,1)

|D| n∈ {1, . . . , N}, D∈ Dⁿh, (3.2) r_Dⁿ := (rⁿ,1)

|D| n∈ {1, . . . , N}, D∈ Dⁿh. (3.3) We next define

Sⁿ(x) := 1 τn

Z tn

t_n−1

S(x, t) dt x∈Ω, n∈ {1, . . . , N}. The elements Sⁿ

D,E,V_D ∈ Vh^n,int,V_E ∈ Vhⁿ, of the finite element diffusion matrix are then given by Sⁿ_D,E :=− X

K∈T_hⁿ

(Sⁿ_h∇ψVE,∇ψVD)K n∈ {1, . . . , N}. (3.4) Here ψVD is the finite element basis function associated withVD ∈ Vhⁿ: the function linear on each K ∈ Thⁿ, equal to 1 atVD, and equal to zero at all otherVE ∈ Vhⁿ. There are two basic choices for the space-time discrete diffusion–dispersion tensor Sⁿ_h: Firstly, we may set

Sⁿ_h|^K := 1

|K|(Sⁿ,1)K n∈ {1, . . . , N}, K ∈ Thⁿ. (3.5)

Note that since ∇ψVD, ∇ψVE are constant on all K ∈ Thⁿ, the above definition is equivalent to directly putting Sⁿ_h =Sⁿ in (3.4). An alternative choice is to define Sⁿ_h by

Sⁿ_h|K:=

1

|K|([Sⁿ]⁻¹,1)_K −1

n∈ {1, . . . , N}, K∈ Thⁿ. (3.6) In fact, the members Sⁿ_D,E forVE ∈ Vh^n,ext do not occur in the definition of the scheme (3.1), but it shows convenient to define them. Finally, we define the value uⁿ_D,E, used for the evaluation of the convective flux across σD,E ∈ Fhⁿ,n∈ {1, . . . , N},D∈ Dhⁿ, by

uⁿ_D,E :=

uⁿ_D+αⁿ_D,E(uⁿ_E−uⁿ_D) if hvⁿ·n,1i^σD,E ≥0

uⁿ_E +αⁿ_D,E(uⁿ_D−uⁿ_E) if hvⁿ·n,1i^σD,E <0 . (3.7) Here αⁿ_D,E is the coefficient of the amount of upstream weighting which is defined by

αⁿ_D,E:=

















 max

min Sⁿ

D,E,¹₂|hvⁿ·n,1iσD,E| ,0

|hvⁿ·n,1i^σD,E| if hvⁿ·n,1i^σD,E 6= 0 andσD,E∈ Fh^n,int

or if σD,E ∈ Fh^n,ext and hvⁿ·n,1i^σD,E >0 0 if hvⁿ·n,1i^σD,E = 0 or ifσD,E∈ Fh^n,ext

and hvⁿ·n,1iσD,E <0

.

The valuesuⁿ_D define, for eachn∈ {1, . . . , N}, the usual piecewise linear (onThⁿ) finite element approximation uⁿ_h by

uⁿ_h := X

D∈D_hⁿ

uⁿ_DψVD. (3.8)

By the approximate solution of problem (1.1a)–(1.1c) by means of the combined finite volume–finite element scheme (3.1), we understand a function u_h,τ such that

u_h,τ(·, tn) :=uⁿ_h n∈ {0, . . . , N},

uh,τ(x,·) is linear on [t_n−1, tn], n∈ {1, . . . , N}, for all x∈Ω. (3.9) Remark 3.1 (Convergence). The convergence of u_h,τ, when the meshes Dhⁿ, Thⁿ do not change with time, towards the weak solution of problem (1.1a)–(1.1c) given by (2.2) has been proved (for a more general nonlinear degenerate case) in [13].

Remark 3.2 (Discrete maximum principle). If Sⁿ_D,E ≥ 0 for all n ∈ {1, . . . , N}, D ∈ Dⁿh, and VE ∈ M(VD) and under suitable conditions on the data, the discrete maximum principle for the combined scheme holds, see [13, Theorem 4.11].

Remark 3.3 (Arithmetic versus harmonic averaging). We remark that the choice (3.5) for Sⁿ_h corresponds to the arithmetic average of the diffusion–dispersion tensor S in space, whereas the choice (3.6) corresponds to the harmonic one.

Remark 3.4 (Relation to the vertex-centered finite volume method). Let the meshesThⁿ be given first and let the meshes Dhⁿ be constructed using the face, edge, and element barycentres. Then the combined finite volume–finite element scheme (3.1) is equivalent to the classical vertex-centered finite volume method (cf., e.g., [19] and the references therein), where mass lumping has been used in the time evolution and reaction terms. This follows easily using [3, Lemma 3] for the diffusion term. Hence, all the results of the present paper apply to the vertex-centered finite volume method as a particular case.

4 A posteriori error estimate and its efficiency

We summarize in this section our a posteriori estimate on the error between the weak solution u and the approximate solution uh,τ and its efficiency.

4.1 Some useful inequalities

Let D⊂R^d be a polygon. The Poincar´e inequality states that

kϕ−ϕDk²D ≤C_P,Dh²_Dk∇ϕk²D ∀ϕ∈H¹(D), (4.1) where ϕD is the mean of ϕover D given by ϕD := (ϕ,1)D/|D| and where hD is the diameter of D. The constant C_P,D can be evaluated as 1/π² if D is convex, cf. [20, 4], and only depends on the geometry of Dif D is nonconvex, cf. [12, Lemma 10.4].

LetD⊂R^dbe a polygon such that |∂Ω∩∂D| 6= 0. Then the Friedrichs inequality states that kϕk²D ≤C_F,D,∂Ωh²_Dk∇ϕk²D ∀ϕ∈H¹(D) such thatϕ= 0 on ∂Ω∩∂D. (4.2) As long as ∂Ω is such that there exists a vector b ∈ R^d such that for almost allx ∈D, the first intersection of B^x and ∂D lies in ∂Ω, where B^x is the straight semi-line defined by the origin x and the vector b, C_F,D,∂Ω = 1, cf. [28, Remark 5.8]. To evaluate C_F,D,∂Ω in the general case is more complicated but it still can be done, cf. [28, Remark 5.9] or [6, Section 3]. As a particular case, whenD= Ω, the Friedrichs inequality reads

kϕk² ≤C_Fh²_Ωk∇ϕk² ∀ϕ∈H₀¹(Ω), (4.3) and C_F = 1.

Finally, for a simplexK ⊂R^d, the trace inequality states that

kϕk²σ ≤C_t,K,σ(h⁻¹_K kϕk²K+kϕk^Kk∇ϕk^K) ∀ϕ∈H¹(K). (4.4) It follows from [22, Lemma 3.12] that the constant Ct,K,σ can be evaluated as |σ|hK/|K|.

4.2 Diffusive and convective flux reconstructions

Let Dⁿh and Thⁿ be the primal and dual meshes as introduced in Section 3.1. For n∈ {2, . . . , N}, letS_h^n−1,n be an additional tertial simplicial mesh, a conforming refinement of allD_hⁿ⁻¹,Dhⁿ,T_hⁿ⁻¹, and Thⁿ, and for n = 1, let Sh^0,1 be a simplicial conforming refinement of both Dh¹ and Th¹. An example of the three types of meshes Dhⁿ,Thⁿ, and Sh^n−1,n is given in Figure2 below. We denote by S^D the partition of D ∈ Dⁿh by the elements of Sh^n−1,n, by K ∈ Sh^n−1,n a generic element of Sh^n−1,n, by G^K its sides, and byGK^int its sides not contained in ∂Ω. By G^D, we denote all the sides of the partitionS^D for a givenD, except those included in the boundary∂Ω. We will also use the obvious notation Sh^n−1,n,int and Sh^n−1,n,ext.

Let X_h denote the space of scalar functions continuous in space and piecewise linear on the meshes Sh^n−1,n and piecewise linear and continuous in time. A function vh ∈ Xh is, as in (3.9), uniquely prescribed by the functions vⁿ_h defined on the meshes Sh^n−1,n, the values of vh at the times t_n, n ∈ {1, . . . , N}, and by the function v⁰_h defined on S_h^0,1, the values of v_h at the time 0. Typically, the values uⁿ_D, D ∈ Dhⁿ, n ∈ {1, . . . , N}, define through (3.8)–(3.9) the function uh,τ ∈Xh. LetVh denote the space of vector functions belonging to the Raviart–Thomas–N´ed´elec spaces of lowest order (cf. [5]) on the meshes Sh^n−1,n in space and piecewise constant in time. A

function u_h ∈ V_h is uniquely prescribed by the functions uⁿ_h defined on the meshes Sh^n−1,n, the values of uh on the time intervals (t_n−1, tn], n∈ {1, . . . , N}. Recall that the distinctive feature is that the normal component of the functionsu_his continuous ([5]); the valuesuⁿ_h·n|σ|=huⁿ_h·n,1iσ

on the sidesσ of the meshes Sh^n−1,n,n∈ {1, . . . , N}, represent the degrees of freedom in the space Vh. Following [29,31,11], our a posteriori error estimates rely on the concept of thediffusive flux reconstructionθ_h∈V_h; following [10], we also introduce aconvective flux reconstructionw_h∈V_h. In order to obtain the upper bounds of Section 4.3below, we merely suppose that there exist values uⁿ_D anduⁿ⁻¹_D and functionsθ_h,w_h ∈V_h such that

uⁿ_D+uⁿ⁻¹_D

τ_n |D|+hθ_hⁿ·n,1i∂D+hwⁿ_h ·n,1i∂D+r_Dⁿuⁿ_D|D|+q_Dⁿ|D|=f_Dⁿ|D|

∀n∈ {1, . . . , N},∀D∈ Dⁿh.

(4.5)

Here r_Dⁿ andf_Dⁿ are given by (3.2)–(3.3). Relation (4.5) is a local conservation property: the term (uⁿ_D +uⁿ⁻¹_D )|D|/τn represents the time accumulation in the volume D,hθⁿ

h ·n,1i∂D the diffusive flux over ∂D, hwⁿ_h ·n,1i∂D the convective flux over ∂D, and r_Dⁿuⁿ_D|D| and f_Dⁿ|D|, respectively, the reaction and sources accumulation in the volume D. Finally, the term qⁿ_D|D| represents a quadrature error when the local conservation is not satisfied exactly for the given scheme. A vast class of locally conservative methods enters the framework of (4.5). In particular, the combined finite volume–finite element scheme (3.1) can be written in the form (4.5) with

hθ_hⁿ·n,1i^σ :=−hSⁿ_h∇uⁿ_h·n,1i^σ σ ⊂∂D, D∈ Dhⁿ (4.6) and

hwⁿ_h·n,1iσD,E :=hvⁿ·n,1iσD,E uⁿ_D,E σ_D,E ∈ FD, D∈ Dhⁿ; (4.7) here, by the notation σ, we mean a side of G^K for some K ∈ S^D which is such that σ ⊂∂D. As the diffusive fluxes hθ_h ·n,1iσ given by (4.6) do not necessarily coincide completely with those stemming from the combined scheme (3.1), we use the additional factorqⁿ_D|D|to fix the balance.

In numerical experiments in Section 7 below, the values q_Dⁿ|D| were negligible. Equations (4.6) and (4.7) do not prescribeθ_h andw_huniquely; they are sufficient, altogether with (4.12) below, for the present theoretical analysis. Practically, there exist several possibilities for fixing the remaining degrees of freedom ofθ_h and wh, following [31].

4.3 A posteriori error estimate

In this section, we suppose that uh,τ ∈ Xh and θ_h,w_h ∈ V_h are arbitrary but such that (4.5) holds. A particular example is the combined finite volume–finite element scheme (3.1) with the construction (4.6)–(4.7). Let n∈ {1, . . . , N} and D∈ Dhⁿ. We set

mⁿ_D := minn

C_P,D¹² h_D(cⁿS,D)⁻¹²,(cⁿv,r,D)⁻¹²o and define the residual estimatorη_R,Dⁿ by

ηⁿ_R,D :=mⁿ_D

f_Dⁿ −uⁿ_D−uⁿ⁻¹_D

τn − ∇ ·θ_hⁿ− ∇ ·wⁿ_h −r_Dⁿuⁿ_D−q_Dⁿ

_D. (4.8)

The diffusive and convective flux estimatorη_DCF,Dⁿ is given by ηⁿ_DCF,D(t) :=

S¹²∇u_h,τ+S⁻¹²θ_hⁿ−S⁻¹²u_h,τv+S⁻¹²w_hⁿ

_D(t). (4.9)

Let K∈ Sh^n−1,n. We then define the cutoff factors mⁿ_K := minn

C

1

P,F,K2 hK(cⁿS,K)⁻¹²,(cⁿv,r,K)⁻¹²o , e

mⁿ_K := minn

(CP,F,K +C

1 2

P,F,K)hK(cⁿS,K)⁻¹, h⁻¹_K (cⁿv,r,K)⁻¹+¹₂(cⁿv,r,K)⁻¹²(cⁿS,K)⁻¹²o ,

where C_P,F,K :=C_P,K forK ∈ Sh^n−1,n,int and C_P,F,K :=C_F,K,∂Ω for K ∈ Sh^n−1,n,ext. We will need below also the diffusive flux estimator η_DF,Dⁿ

η_DF,Dⁿ (t) :=

( X

K∈SD

mⁿ_Kk∇ ·(S∇u_h,τ+θⁿ_h)kK

+ (meⁿ_K)¹² X

σ∈G^int_K

(C_t,K,σ)¹²k(S∇u_h,τ+θⁿ_h)·nkσ

!2)¹₂ (t).

Lastly, set

mⁿ:= minn C

1 2

Fh_Ω(cⁿS,Ω)⁻¹²,(cⁿv,r,Ω)⁻¹²o . Then the data oscillation–quadrature estimator η_DOQ,Dⁿ is given by

η_DOQ,Dⁿ (t) :=mⁿ

f−f_Dⁿ−(uh,τ)t+uⁿ_D −uⁿ⁻¹_D

τn −ruh,τ+rⁿ_Duⁿ_D+qⁿ_D

D

(t). (4.10) We finally denote

η^(1),n2

:=

Z tn

tn−1

( X

D∈Dⁿ_h

η_R,Dⁿ +ηⁿ_DCF,D(t)2

)¹₂ +

( X

D∈Dⁿ_h

η_DOQ,Dⁿ (t)2

)¹₂!2

dt,

η^(2),n2

:=

Z tn

tn−1

( X

D∈Dⁿ_h

η_R,Dⁿ 2

)¹₂ +

( X

D∈Dⁿ_h

η_DF,Dⁿ (t)2

)¹₂

+ sup

ϕ∈H₀¹(Ω);|||ϕ|||=1

(∇ ·(uh,τv−wⁿ_h), ϕ) + ( X

D∈Dⁿ_h

η_DOQ,Dⁿ (t)2

)¹₂!2

dt

and

ηⁿ:= min

η^(1),n, η^(2),n , (4.11a)

η:=

( _N X

n=1

(ηⁿ)² )¹₂

. (4.11b)

Remark 4.1 (Data oscillation–quadrature estimators). In a posteriori estimates for finite ele- ment, vertex-centered finite volume, or mixed finite element methods, cf. [25, 11], data oscillation estimators of the form mⁿkf −f_Dⁿk appear. The other terms of the present estimators η_DOQ,Dⁿ are related to the non-variational general nature of (4.5) (or of (3.1)). They clearly represent a numerical quadrature error, which is illustrated below on numerical experiments. If the vertex- centered finite volume method, cf., e.g. [31], was used instead of (4.5), then ηⁿ_DOQ,D would reduce to the usual data oscillation estimator mⁿkf −f_Dⁿk.

Remark 4.2 (Estimatorsη^(1),n and η^(2),n). The estimator η^(1),n is fully computable and is based on the constant-free estimator (4.9). The estimator η^(1),n is, however, not robust with respect to the convection dominance, as noted in [8, 27]. Using the estimator η^(2),n, on the contrary, robust lower bound will be proven below following [25]. The estimatorη^(2),n is, however, not computable.

Its computable upper bound (still ensuring robustness) can be established following [25, Section 8], for the price of a solution of an auxiliary stationary reaction–diffusion problem on each time step.

Such an approach is not pursued here in view of its complexity.

Using the above definitions, we will prove in Section 5.1below the following a posteriori error estimate:

Theorem 4.3 (Energy norm a posteriori error estimate). Let u be the weak solution of prob- lem (1.1a)–(1.1c) given by (2.2)and letuh,τ ∈Xh andθ_h,wh∈Vh be arbitrary but such that (4.5) holds. Then

k(u−uh,τ)(·, T)k²+ku−uh,τk²X ≤η²+ku₀−uh,τ(·,0)k².

Theorem 4.4(Augmented norm a posteriori error estimate). Let the assumptions of Theorem4.3 hold. Then

3k(u−uh,τ)(·, T)k²+ku−uh,τk²Y ≤5η²+ 3ku₀−uh,τ(·,0)k². 4.4 Distinguishing the space and time error contributions

The estimate η^(1),n of (4.11a) is fully computable and allows to control the overall error. There is, however, no distinction between the space and time errors. Using the triangle inequality, we have, as in [11], for the example ofη_DCF,Dⁿ (t),

η_DCF,Dⁿ (t)≤

S¹²∇uⁿ_h+S⁻¹²θⁿ_h −S⁻¹²uⁿ_hvⁿ+S⁻¹²w_hⁿ D(t) +

S¹²∇u_h,τ−S¹²∇uⁿ_h−S⁻¹²u_h,τv+S⁻¹²uⁿ_hvⁿ D(t).

Note that wheneverSis constant in time on (t_n−1, tn], the first of the above terms gets independent of time; it clearly corresponds to the spatial part of the error. The second of the above terms then corresponds to the temporal part of the error, so that we have the usual space-time contributions division as in [21,25]. Note as well that when bothSand v are constant in time on (t_n−1, tn], we get, using the definition of uh,τ by (3.9) as in [24, Equation (6.5)], the following easily computable upper bound on Rtn

t_n−1 η_DCF,Dⁿ (t)2

dt ¹²: (Z tn

tn−1

ηⁿ_DCF,D(t)2

dt )¹₂

≤τ

1

n2

(Sⁿ)¹²∇uⁿ_h+ (Sⁿ)⁻¹²θ_hⁿ−(Sⁿ)⁻¹²uⁿ_hvⁿ+ (Sⁿ)⁻¹²wⁿ_h

D

+τn

3 ¹₂

(Sⁿ)¹²∇(uⁿ⁻¹_h −uⁿ_h)−(Sⁿ)⁻¹²vⁿ(uⁿ⁻¹_h −uⁿ_h) _D. We now proceed similarly for the other estimators, for general S and v. We obtain

ηⁿ≤η_spⁿ +ηⁿ_tm

with ηⁿ_sp:=

( τn

X

D∈Dⁿ_h

(η_R,Dⁿ )² )¹₂

+ (Z tn

tn−1

X

D∈D_hⁿ

S¹²∇uⁿ_h+S⁻¹²θ_hⁿ−S⁻¹²uⁿ_hvⁿ+S⁻¹²wⁿ_h²

D(t) dt )¹₂

+ (Z tn

tn−1

X

D∈D_hⁿ

(mⁿ)²

f−f_h−(u_h,τ)_t+ uⁿ_D−uⁿ⁻¹_D

τn −rⁿuⁿ_h+rⁿ_Duⁿ_D+qⁿ_D

2 D

(t) dt )¹₂

and

η_tmⁿ :=

(Z tn

t_n−1

X

D∈D_hⁿ

S¹²∇uh,τ−S¹²∇uⁿ_h−S⁻¹²uh,τv+S⁻¹²uⁿ_hv^{n 2}

D(t) dt )¹₂

+ (Z tn

tn−1

X

D∈D_hⁿ

(mⁿ)²

fh−f_Dⁿ−ruh,τ+rⁿuⁿ_h ²

D(t) dt )¹₂

. Consequently, we have the following corollary:

Corollary 4.5(Energy norm a posteriori error estimate distinguishing the space and time errors).

Let the assumptions of Theorem 4.3hold. Then

k(u−u_h,τ)(·, T)k²+ku−u_h,τk²X ≤ XN n=1

(η_spⁿ +η_tmⁿ )²+k(u−u_h,τ)(·,0)k².

4.5 Efficiency of the estimate

Henceforth, let a.b denotea≤Cb for a generic positive constantC, not necessarily the same at each occurrence, dependent on the shape regularity of the meshes Sh^n−1,n and dbut independent of the space domain Ω, final time T, and the parameters S, v, r, f, and u0; C can only depend on the local oscillation of these parameters. Similarly, the notation a ≈ b will be employed when simultaneously a.b and b.a. From now on, we also omit denoting explicitly by (t) the dependence of the estimators on the time. We will suppose below that the followingapproximation property of the diffusive flux reconstruction θ_h is satisfied:

kSⁿ_h∇uⁿ_h +θ_hⁿk^D . (X

σ∈GD

hσk[[Sⁿ_h∇uⁿ_h·n]]k²σ

)¹₂

∀n∈ {1, . . . , N},∀D∈ Dⁿh. (4.12) Here [[·]] denotes the jump across a side. This property can be shown, in dependence on the construction ofθⁿ

h from∇uⁿ_h, as in [31]. We also introduce some additional notation. Letf_h, r_h ∈ Xhandv_h ∈V_hbe space-time discrete approximations of the source term, of the reaction function, and of the velocity field v, respectively. Let n ∈ {1, . . . , N} and D ∈ Dⁿh. We define the data oscillation estimator η_DO,Dⁿ by

ηⁿ_DO,D :=mⁿ_Dkf_Dⁿ−f_hⁿkD+mⁿ_Dkuⁿ_h(r_hⁿ−r_Dⁿ)kD

+ ( X

K∈SD

(mⁿ_K)²k∇ ·(S−Sⁿ_h)∇u_h,τk²K+meⁿ_K X

σ∈G_K^int

k(S−Sⁿ_h)∇u_h,τ·nk²σ

!)¹₂

+ ( X

K∈SD

(mⁿ_K)²k∇ ·((v−vⁿ_h)u_h,τ)k²K+meⁿ_K X

σ∈G^int_K

k(v−vⁿ_h)u_h,τ ·nk²σ

!)¹₂

and thequadrature estimator η_Q,Dⁿ by ηⁿ_Q,D:=mⁿ_D

uⁿ_h−uⁿ⁻¹_h

τ_n −uⁿ_D−uⁿ⁻¹_D

τ_n +r_Dⁿ(uⁿ_h−uⁿ_D)−qⁿ_D

D

+ ( X

K∈SD

(mⁿ_K)²k∇ ·(vⁿ_huⁿ_h −wⁿ_h)k²K +meⁿ_K X

σ∈G_K^int

k(vⁿ_huⁿ_h−wⁿ_h)·nk²σ

!)¹₂ .

Finally, for the sake of simplicity, we make in this section an additional assumption that∇ ·v= 0.

In this case, the augmented norm (2.7) is closely related to that of [25], so that we can use the results of [25] for the lower bound proof. We then have the following theorem:

Theorem 4.6(Efficiency of the a posteriori error estimators in the augmented norm). Let∇·v= 0, letube the weak solution of problem(1.1a)–(1.1c) given by(2.2)and letuh,τ ∈Xh andθ_h,w_h ∈V_h be arbitrary but such that (4.12) holds. Let n ∈ {1, . . . , N}, let Sh^n−1,n be shape-regular, and let, for all D∈ Dⁿh,hD ≈hK for allK ∈ S^D. Then

(ηⁿ)² .ku−uh,τk²Y(tn−1,tn)+ Z tn

tn−1

X

D∈Dⁿ_h

{(η_DOQ,Dⁿ )²+ (ηⁿ_DO,D)²+ (η_Q,Dⁿ )²}dt +kf −fⁿ−(r−rⁿ)uh,τk²X^′(tn−1,tn).

5 Proof of the a posteriori error estimate and of its efficiency

We shall prove in this section the a posteriori error estimates stated in Theorems 4.3 and 4.4, as well as their efficiency discussed in Theorem 4.6.

5.1 Proof of the a posteriori error estimate Lemma 5.1 (Bound by the residual). There holds

k(u−uh,τ)(·, T)k²+ku−uh,τk²X ≤ sup

ϕ∈X,kϕkX=1

Z T

0 hB(u−uh,τ), ϕidt

!2

+k(u−uh,τ)(·,0)k². Proof. Using the obvious relation

Z _T

0 hB(v), vidt≤ sup

ϕ∈X,kϕkX=1

Z _T

0 hB(v), ϕidtkvkX ≤ 1

2 sup

ϕ∈X,kϕkX=1

Z _T

0 hB(v), ϕidt

!2

+1 2kvk²X, valid for any v∈Z, the assertion follows from (2.4) and the fact that (u−uh,τ)∈Z.

Let from now on, ϕ ∈ X with kϕk^X = 1 be fixed. Then, on the time interval (t_n−1, tn], n∈ {1, . . . , N}, we have the following results.

Lemma 5.2 (Equivalent form of the residual). There holds

hB(u−uh,τ), ϕi=TR(ϕ) +TDCF(ϕ) +TQ(ϕ)