A posteriori error estimates for a fully discrete approximation of Sobolev equations

CONFLUENTES MATHEMATICI

Serge NICAISE and Fatiha BEKKOUCHE

A posteriori error estimates for a fully discrete approximation of Sobolev equations Tome 11, n^o1 (2019), p. 3-28.

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11, 1 (2019) 3-28

A POSTERIORI ERROR ESTIMATES FOR A FULLY DISCRETE APPROXIMATION OF SOBOLEV EQUATIONS

SERGE NICAISE AND FATIHA BEKKOUCHE

Abstract. The paper presents an a posteriori error estimator for a (piecewise linear) conforming finite element approximation of some (linear) Sobolev equations in R^d,d= 2 or 3, using implicit Euler’s scheme. For this discretization, we derive a residual indicator, which uses a spatial residual indicator based on the jumps of conormal derivatives of the approximations and a time residual indicator based on the jump (in an appropriated norm) of the successive solutions at each time step. Lower and upper bounds are obtained with minimal assumptions on the meshes. Numerical experiments that confirm and illustrate the theoretical results are given.

1. Introduction

This paper deals with the a posteriori analysis of linear Sobolev equations of type

L₁ut+L₂u=f in Ω×(0, T),

where L₁, L₂ are second order differential operators, approximated using implicit Euler’s scheme in time and a (piecewise linear) conforming finite element approx- imation in space. Such problems are interesting not only because they are gener- alizations of a standard parabolic problem but also because they arise naturally in a large variety of applications (model of fluid flow in fissured porous media [2], two-phase flow in porous media with dynamical capillary pressure [12, 14], heat conduction in two-temperature systems [8, 25] and shear in second order fluids [11, 24]).

Several approaches have been introduced to define error estimators for parabolic problems (like the heat equation, corresponding to the case whereL1 is reduced to the identity operator), let us quote [4, 5, 6, 7, 15, 18, 21, 22, 27, 28, 29]. To be able to extend these techniques to Sobolev equations, we need to be able to manage the replacement of the identity operator by a second order elliptic one. To the best of our knowledge such an approach has not been considered. Indeed we only found two papers related to this topic. The first one [17] highlights a superconvergence phenomena on cartesian grids whose estimates can be bounded by the norms of known data so that some useful a posteriori error estimates can be derived, while the second one [26] obtains some error estimates by solving local nonlinear or linear pseudo-parabolic equations for corrections to the solution.

The schedule of the paper is the following one: Section 2 recalls the continuous problem and its discretizations. In Section 3 we introduce some notations and give some useful properties. Section 4 is devoted to the a posteriori analysis of the time discretization. The efficiency and reliability of the spatial error estimator are established in Section 5. The a posteriori analysis of the full discrete problem is considered in Section 6, where we show the efficiency and reliability of the sum of

Math. classification:65M15, 65M50, 65M60.

Keywords: Sobolev equations, a posteriori error analysis.

3

the spatial and time error estimators. Finally Section 7 is devoted to numerical tests which confirm our theoretical analysis.

Let us finish this section with some notations used in the remainder of the paper.

For a bounded domain D, the usual norm and semi-norm of H^s(D) (s > 0) are denoted by k · ks,D and | · |s,D, respectively. For s = 0, we will drop the index s. Furthermore, the inner product in L²(Ω) will be denoted by (·,·). Finally, the notationA.B(resp.A&B) means the existence of a positive constantC₁(resp.

C₂), which is independent of A and B as well as the discretization parameters h and τ_p such that A 6 C₁B (resp.A > C₂B). The notation A ∼B means that A.B andA&B hold simultaneously.

2. The continuous, time semi-discrete and full discrete problems Let Ω be an open bounded of R^d, d = 2 or 3, with a polygonal (d = 2) or polyhedral (d= 3) boundary Γ. LetT be a positive and fixed real number.

Fori= 1,2, letLibe a second order elliptic operator in the form Li(x, Dx)u=−

d

X

k,`=1

∂k(a⁽ⁱ⁾<sub>k,`</sub>(x)∂`u) +

d

X

k=1

b⁽ⁱ⁾_k (x)∂ku+c⁽ⁱ⁾(x)u, wherea⁽ⁱ⁾_{k,`</sub>=a<sup>(i)</sup><sub>`,k}, b⁽ⁱ⁾_k , c⁽ⁱ⁾∈L^∞(Ω) and introduce the bilinear forms onH₀¹(Ω)

ai(u, v) = Z

Ω





d

X

k,`=1

a⁽ⁱ⁾<sub>k,`</sub>(x)∂`u ∂kv+

d

X

k=1

b⁽ⁱ⁾_k (x)∂ku v+c⁽ⁱ⁾(x)u v



dx.

We suppose thata1 anda2 are symmetric, thata2 is non negative, i.e.,

a2(u, u)>0 for allu∈H₀¹(Ω), (2.1) and that a1is coercive in H₀¹(Ω), namely there existsα >0 such that

a1(u, u)>αkuk²_1,Ω for allu∈H₀¹(Ω). (2.2) In this setting, we consider the following Sobolev equation: Letube the solution of















L₁u_t+L₂u=f in Ω×(0, T), u(·, t) = 0 on Γ×(0, T), u(·,0) =u0 in Ω,

(2.3)

where ut means the time derivative of u. The datum f is supposed to satisfy f ∈L²(0, T;H⁻¹(Ω)) and the initial valueu0∈H₀¹(Ω). Under these assumptions, problem (2.3) or equivalently

a1(ut(t), v) +a2(u(t), v) = (f(t), v) for allv∈H₀¹(Ω) and a.e.t∈(0, T) (2.4) has a unique (weak) solution in C([0, T];H₀¹(Ω)), see [3]. This system is a linear Sobolev equation in Ω, where some a priori error analyses were performed in [1, 13, 17, 19, 20, 23] in some particular situations or with a kind of Neumann boundary conditions. Some a posteriori error analyses can be found in [17, 26].

Without loss of generality, we can assume that a2 is also coercive in H₀¹(Ω), indeed by the change of unknown

u(·, t) =˜ e^−λtu(·, t),

for a positive real numberλ, we see that (2.4) is equivalent to

a1(˜ut(t), v) + ˜a2(˜u(t), v) = (e^−λtf(t), v) for allv∈H₀¹(Ω) and a.e.t∈(0, T), (2.5) where

˜a₂(u, v) =a₂(u, v) +λa₁(u, v) for allu, v∈H₀¹(Ω)

is clearly coercive inH₀¹(Ω) due to (2.1)-(2.2). Hence from now on we also suppose that a2 is coercive and denote by

kuka_i =p

ai(u, u), for allu∈H₀¹(Ω),

two equivalent norms ofH₀¹(Ω). We further denote bykuk₋₁ the norm inH⁻¹(Ω) obtained by using the duality with the second norm ofH₀¹(Ω),in other words,

kgk₋₁= sup

v∈H¹₀(Ω),v6=0

|hg;vi|

kvka2

, for allg∈H⁻¹(Ω), whereh·;·imeans the duality pairing betweenH⁻¹(Ω) and H₀¹(Ω).

2.1. Time discretization using implicit Euler’s scheme. We now suppose that f ∈C([0, T];H⁻¹(Ω)). We further introduce a partition of [0, T] into subin- tervals [t_p−1, t_p], 1 6p6 N such that 0 = t₀ < t₁ < · · · < t_N = T. Denote by τ_p=t_p−t_p−1 the length of [t_p−1, t_p] and byτ= max_pτ_pthe global time mesh size.

The semi-discrete approximation of the continuous problem (2.3) by an implicit Euler scheme consists in finding a sequence (u^p)_06p6N solution of

















 L₁

u^p−u^p−1 τ_p

+L₂u^p=f^p in Ω, 16p6N,

u^p= 0 on Γ, 16p6N,

u⁰=u0 in Ω,

(2.6)

withf^p=f(·, tp). This problem admits a unique weak solutionu^p∈H₀¹(Ω), whose variational formulation is

a₁(u^p−u^p−1 τp

, v) +a₂(u^p, v) = Z

Ω

f^pv, for allv∈H₀¹(Ω), (2.7) or equivalently

a₁(u^p, v) +τ_pa₂(u^p, v) =a₁(u^p−1, v) +τ_p Z

Ω

f^pv, for allv∈H₀¹(Ω). (2.8) The unique solvability of the variational formulation (2.8) is clearly a direct conse- quence of the Lax-Milgram lemma.

Remark 2.1. — An a priori error analysis of the explicit Euler scheme a1(u^p−u^p−1

τp

, v) +a2(u^p−1, v) = Z

Ω

f^p−1v, for allv∈H₀¹(Ω), (2.9) was considered in [3] since it is more appropriate for Sobolev equations. The a posteriori error analysis that we perform below for system (2.8) is immediately applicable to (2.9) since it can be re-written as

a1,p(u^p−u^p−1 τp

, v) +a2(u^p, v) = Z

Ω

f^p−1v, for allv∈H₀¹(Ω), where

a_1,p(u, v) =a₁(u, v)−τ_pa₂(u, v), that is coercive uniformly inp, ifτ_p is small enough.

2.2. Full discretization. Problem (2.8) is now discretized by a conforming finite element method. For that purpose, for anyp= 0,1,· · · , N, let us fix a conforming mesh T_ph of Ω which is regular in Ciarlet’s sense [9, p. 124]. All elements are triangles or tetrahedra and will be denoted by K. For an element K ∈ T_ph, we recall that h_K is the diameter of K and that h_p = max

K∈T_phh_K. The set of all edges/faces ofTph is denoted byEph. LetE_ph^int be the set of interior edges/faces of TphandEK be the set of the edges/faces of the elementK. Finally for an edge/face E∈ EK∩ EL we denote byhE= ^d₂(^|K|_|E|+^|L|_|E|), its mean height.

Introduce the conforming finite element space:

Vph={v∈H₀¹(Ω) :v_|K ∈P1, for allK∈Tph}.

The fully discrete approximation of problem (2.3) using Euler’s scheme and the conforming finite element is then given by: Given an approximation u⁰_h ∈ V0h of u0, findu^p_h∈Vph, 16p6N, such that

a₁(u^p_h−u^p−1_h τp

, v_h) +a₂(u^p_h, v_h) = Z

Ω

f^pv_h, for allv_h∈V_ph, (2.10) or equivalently

a1(u^p_h, vh) +τpa2(u^p_h, vh) =a1(u^p−1_h , vh) +τp

Z

Ω

f^pvh, for allvh∈Vph. (2.11) Definition 2.2. — Let u^p be a solution of (2.8) and u^p_h a solution of (2.11), then we denote the spatial error by

e^p=u^p−u^p_h.

3. Some useful notations and properties

For a boundary edge/faceEwe denote the outward normal vector byn_E. Given an interior edge/faceE, we choose an arbitrary normal directionn_Eand denote by K_inandK_extthe two elements sharing this edge/face. Without any restriction, we may suppose here that nE is pointing toKextlike in Figure 3.1.

The jump of a functionv across an edge/faceE at a pointxis defined by

v(x)

E=

( lim

α→0⁺(v(x+αnE)−v(x−αnE) ifE∈ E_ph^int,

v(x) ifE∈ E_ph\ E_ph^int.

@

@

@

@XXXXXXXXXXXXXXX -

n

K

K

Figure 3.1. Two elements sharing the edgeE Note that the sign of

v(x)

Edepends on the orientation ofnE. However, quantity like a gradient jump

∇v·n_E

E is independent of this orientation.

In the sequel we will use local patches: for an element K we define ω_K as the union of all elements having a common edge/face withK, for an edge/faceE, let ωE be the union of both elements havingE as an edge/face and finally for a node x, letωx be the union of all elements havingxas a node. Similarly denote by ˜ωK

(resp. ˜ωE) the union of all triangles sharing a node withK (resp.E).

Recall that the Clément interpolation operator is defined as follows: Denote by Nph the set of nodes of the triangulationTph and byN_ph^int the set of interior nodes of the triangulationTph. For each nodex∈ N_ph^intdenote further byλxthe standard hat function associated withx, namelyλx∈Vph and satifies

λx(y) =δx,y, for ally∈ N_ph^int. For anyw∈L²(Ω), we defineI_C⁰wby

I_C⁰w= X

x∈N_ph^int

|ωx|⁻¹Z

ω_x

w

λx. (3.1)

Note thatI_C⁰wbelongs toVph. Moreover this operator has the following properties [10]:

Lemma 3.1. — For allw∈H₀¹(Ω), we have

kw−I_C⁰wkK .hKk∇wkω˜_K, for allK∈Tph, (3.2) kw−I_C⁰wkE.h^1/2_E k∇wkω˜_E, for allE∈ E_ph^int, (3.3) k∇I_C⁰wkK .k∇wkω˜_K, for allK∈Tph. (3.4) IfK∈Tph, then the element residual is defined onK by

R^p_K =

f(·, tp)−L1

u^p_h−u^p−1_h τp

−L2u^p_h

|K, while ifE∈ E_ph^int, then the edge/face residual is

J_E,n^p =

A1∇_up h−u^p−1_h

τ_p

+A2∇u^p_h

·nE

E

, where fori= 1 or 2,Ai is thed×dsymmetric matrix given by

Ai= (a⁽ⁱ⁾<sub>k,`</sub>)16k,`6d.

Now we prove a property satisfied by the spatial errore^p that we will use in the proof of the spatial error bounds.

Lemma 3.2(Galerkin orthogonality). — The errore^p satisfies the Galerkin or- thogonality relation

a₁(e^p−e^p−1 τp

, v_h) +a₂(e^p, v_h) = 0, for allv_h∈V_ph. (3.5) Proof. — It suffices to subtract (2.7) withv=vh∈Vphto the identity (2.10).

Lemma 3.3. — The errore^p satisfies a₁(e^p−e^p−1

τ_p , v) +a₂(e^p, v) = X

K∈Tph

Z

K

R^p_Kv+ X

E∈E_ph^int

Z

E

J_E,n^p v, for allv∈H₀¹(Ω).

(3.6) Proof. — We first observe that

a1(e^p−e^p−1

τ_p , v) +a2(e^p, v) =a1(u^p−u^p−1

τ_p , v) +a2(u^p, v)

− a1(u^p_h−u^p−1_h τp

, v) +a2(u^p_h, v)

! . We transform the first term on the right-hand side using (2.7) and the second one by elementwise integration by parts, reminding that

ai(u, v) = X

K∈T_ph

Z

K

(Ai∇u· ∇v+

d

X

k=1

b⁽ⁱ⁾_k ∂kuv+c⁽ⁱ⁾uv

! .

This leads to the conclusion.

The above lemmas allow us to prove the following lemma.

Lemma 3.4. — The following identity holds a1(e^p, e^p) +τpa2(e^p, e^p) =a1(e^p−1, e^p) +τp

X

K∈Tph

Z

K

R^p_K(e^p−I_C⁰e^p) +τ_p X

E∈E_ph^int

Z

E

J_E,n^p (e^p−I_C⁰e^p). (3.7) Proof. — We write

a₂(e^p, e^p) =a₂(e^p, e^p−I_C⁰e^p) +a₂(e^p, I_C⁰e^p),

then we transform the first term using (3.6) with v = e^p−I_C⁰e^p and the second term using the Galerkin orthogonality relation (3.5) withv_h=I_C⁰e^p.

4. A posteriori analysis of the time discretization

Inspired from [4, 6, 16, 18, 21], that considered the heat equation, we define the time error indicator by

η^p_t =τ_p^1/2ku^p_h−u^p−1_h ka2,16p6N. (4.1) The only difference with the above papers lies on the chosen norm ofu^p_h−u^p−1_h .

For shortness we introduce the following notation: Denote byπ_τf the step func- tion which is constant and equal tof(t_p) on each interval (t_p−1, t_p), 16p6N. For

a sequence v^p ∈ H₀¹(Ω), 06 p6N, we denote by vτ its “Lagrange" interpolant, which is affine on each interval [tp−1, tp], 1 6p6 N, and equal to v^p at tp, i.e., defined by,

vτ(t) = tp−t τp

v^p−1+t−tp−1

τp

v^p, for allt∈[tp−1, tp],16p6N.

Denote finallye_τ=u−u_τ, the time discretization error.

As

∂tuτ =u^p−u^p−1

τ_p on (t_p−1, tp), the semi-discrete equation (2.7) is equivalent to

a₁(∂_tu_τ(t), v) +a₂(u^p, v) = (f^p, v), for allv∈H₀¹(Ω) and allt∈(t_p−1, t_p). (4.2) Taking the difference with (2.4), we derive the residual equation

a1(∂teτ(t), v) +a2(eτ(t), v) = ((f−πτf)(t), v) +a2((u^p−uτ)(t), v), (4.3) for allv∈H₀¹(Ω) and a.e.t∈(t_p−1, tp).

This identity allows us to prove the next error bound.

Theorem 4.1 (Time upper error bound). — The next estimate holds ke_τ(t_n)k²_a

1+ Z tn

0

ke_τ(s)k²_a

2ds. (4.4)

n

X

p=1

(η^p_t)²+ Z t_n

0

k(uτ−u_hτ)(s)k²_a2ds+kf−π_τfk²_L2(0,tn;H⁻¹(Ω)). Proof. — The residual equation (4.3) withv=eτ(t) yields

a1(∂teτ(t), eτ(t)) +a2(eτ(t), eτ(t)) = ((f−πτf)(t), eτ(t)) +a2((u^p−uτ)(t), eτ(t)), for a.e.t∈(t_p−1, tp). Asa1 is symmetric, we have

a1(∂teτ(t), eτ(t)) = 1

2∂ta1(eτ(t), eτ(t)), hence integrating the above identity int∈(t_p−1, tp), one gets

1

2a1(eτ(tp), eτ(tp))−1

2a1(eτ(t_p−1), eτ(t_p−1)) + Z t_p

tp−1

a2(eτ(t), eτ(t))

= Z t_p

tp−1

((f−πτf)(t), eτ(t))dt+ Z t_p

tp−1

a2((u^p−uτ)(t), eτ(t))dt.

Using Young’s inequality, one obtains 1

2a1(eτ(tp), eτ(tp))−1

2a1(eτ(t_p−1), eτ(t_p−1)) +1 2

Z t_p tp−1

a2(eτ(t), eτ(t)) (4.5) 6

Z t_p tp−1

k(f−πτf)(t)k²₋₁dt+ Z t_p

tp−1

k(u^p−uτ)(t)k²_a2dt.

We now estimate the second term of this right-hand side. First by the definition ofuτ we clearly have

Z t_p tp−1

k(u^p−uτ)(t)k²_a2dt=τp

3ku^p−u^p−1k²_a2. (4.6)

Secondly, using the triangular inequality, we simply write

τ_p^1/2ku^p−u^p−1ka₂ 6η_t^p+τ_p^1/2ku^p−u^p_hka₂+τ_p^1/2ku^p−1_h −u^p−1ka₂. (4.7) Let us now show that

τ_pku^p−u^p_hk²_a2+τ_pku^p−1−u^p−1_h k²_a2 66 Z tp

tp−1

k(uτ−u_hτ)(t)k²_a2dt. (4.8) Indeed by definition, we have

(uτ−uhτ)(t) =t_p−t τp

(u^p−1−u^p−1_h ) +t−t_p−1 τp

(u^p−u^p_h), for allt∈[tp−1, tp], and therefore

k(u_τ−u_hτ)(t)k²_a

2 =

t_p−t τp

²

ku^p−1−u^p−1_h k²_a

2+

t−t_p−1 τp

²

ku^p−u^p_hk²_a

2

+ 2(tp−t)(t−t_p−1)

τ_p² a2(u^p−1−u^p−1_h , u^p−u^p_h), for allt∈(t_p−1, t_p).

Integrating this expression in t ∈(t_p−1, t_p), one finds after simple calculations that

Z tp

tp−1

k(uτ−u_hτ)(t)k²_a2dt= τ_p

3(ku^p−1−u^p−1_h k²_a2+ku^p−u^p_hk²_a2+a₂(u^p−1−u^p−1_h , u^p−u^p_h)).

Cauchy-Schwarz’s inequality allows to conclude that (4.8) holds.

In conclusion, the identity (4.6) and the estimates (4.7)-(4.8) yield Z t_p

tp−1

k(u^p−uτ)(t)k²_a2dt.(η_t^p)²+ Z t_p

tp−1

k(uτ−uhτ)(t)k²_a2dt. (4.9) This estimate in (4.5) leads to

a₁(e_τ(t_p), e_τ(t_p))−a₁(e_τ(t_p−1), e_τ(t_p−1)) + Z t_p

tp−1

a₂(e_τ(t), e_τ(t)) .(η_t^p)²+

Z tp

tp−1

k(f−πτf)(t)k²₋₁dt+ Z tp

tp−1

k(uτ−uhτ)(t)k²_a

2dt.

Summing this estimate inp= 1,· · ·, nleads to the conclusion.

Corollary 4.2(Second time upper error bound). — The next estimate holds k∂_te_τk²_L2(0,tn;H₀¹(Ω)).

n

X

p=1

(η^p_t)²+ Z tn

0

k(u_τ−u_hτ)(s)k²_a

2ds+kf−π_τfk²_L2(0,t_n;H⁻¹(Ω)). (4.10) Proof. — The residual equation (4.3) and the equivalence between the norms k · ka1 andk · ka2 directly give

k∂teτ(t)ka₁ .k(f−πτf)(t)k₋₁+keτ(t)ka₂+k(u^p−uτ)(t)ka₂, for allt∈(t_p−1, t_p).

Integrating the square of this estimate in t ∈(t_p−1, tp) and summing onp, we obtain

Z t_n 0

k∂teτ(t)k²_a1dt. Z t_n

0

k(f−πτf)(t)k²₋₁dt+ Z t_n

0

keτ(t)k²_a2dt+ Z t_n

0

k(u^p−uτ)(t)k²dt.

The second term of this right-hand side is estimated in (4.4), while the third term

is estimated via (4.9).

Remark 4.3. — In the implementation point of view, all the terms of the right- hand side of (4.4) and of (4.10) should be computable. This is indeed the case for the terms (η_t^p)²andkf−πτfk²_L2(0,tn;H⁻¹(Ω)), while the termRt_n

0 k(uτ−uhτ)(s)k²_a2ds is not, because the exact solutions u^p are used, but it will be estimated by com- putational quantities in the next section (see Theorem 5.2 and the estimate (6.3) below).

Let us go on with the local time lower bound.

Theorem 4.4 (Time lower error bound). — For all p = 1,· · ·, N, the next estimate holds

(η^p_t)². Z t_p

tp−1

(keτ(t)k²_a2+k∂te_τ(t)k²_a1)dt (4.11) +τp(ku^p−u^p_hk²_a2+ku^p−1−u^p−1_h k²_a2) +kf−πτfk²_L2(tp−1,t_p;H⁻¹(Ω)). Proof. — By the triangular inequality we may write

η_t^p.τ_p^1/2(ku^p−u^p−1k_a2+ku^p−u^p_hk_a2+ku^p−1−u^p−1_h )k_a2). (4.12) Hence it remains to estimate the termτp^1/2ku^p−u^p−1ka₂. First we recall the identity (4.6)

τ_p

3 ku^p−u^p−1k²_a

2= Z tp

tp−1

k(u^p−u_τ)(t)k²_a

2dt.

Second taking as test function in (4.3) v = (u^p−uτ)(t), with t ∈ (t_p−1, tp), one obtains

a₁(∂_te_τ(t),(u^p−u_τ)(t)) +a₂(e_τ(t),(u^p−u_τ)(t)) =

((f−πτf)(t),(u^p−uτ)(t)) +a2((u^p−uτ)(t),(u^p−uτ)(t)), for a.e.t∈(t_p−1, tp).

With the help of Cauchy-Schwarz’s inequality and the continuity ofa1 and the coerciveness ofa2, we arrive at

k(u^p−u_τ)(t)k²_a2.k∂te_τ(t)k²_a1+keτ(t)k²_a2+k(f−π_τf)(t)k²₋₁,

for all a.e.t∈(t_p−1, tp). Integrating this estimate in t∈(t_p−1, tp) we deduce that τ_p

3ku^p−u^p−1k²_a2 . Z t_p

tp−1

(keτ(t)k²_a2+k∂teτ(t)k²_a1+kf−πτfk²₋₁)dt.

The conclusion follows by inserting this estimate in (4.12).

5. A posteriori analysis of the spatial discretization

5.1. An upper error bound. As usual [27] the exact element residual R_K^p it is replaced by an approximate element residual

r^p_K =

(f_h^p−L₁ u^p_h−u^p−1_h τp

−L₂u^p_h

|K, (5.1)

where f_h^p is a finite dimensional approximation of f(·, t_p). A possible choice is (f_h^p)_|K := 1

|K|

Z

K

f(x, tp)dx, for allK∈Tph.

Definition 5.1. — Letp>1. The local error estimator η_K^p is defined by η^p_K=h_Kkr^p_Kk_K+ X

E∈EK

h^1/2_E J_E,n^p

_E, while the global oneη^p is given by

(η^p)²= X

K∈Tph

(η_K^p)². The local and global approximation terms are defined by

ξ_K^p =hKkf(·, tp)−f_h^pkω_K, (ξ^p)²= X

K∈T_ph

(ξ^p_K)². Theorem 5.2 (Upper error bound). — The next estimate holds

keⁿk²_a

1+

n

X

p=1

τ_pke^pk²_a

2 .

n

X

p=1

τ_p((η^p)²+ (ξ^p)²) +ke⁰k²_a

1. (5.2)

Proof. — This upper bound is a consequence of Lemma 3.4 by estimating appro- priately each term of the right-hand side of the identity (3.7). First we transform

τp

X

K∈T_ph

Z

K

R^p_K(e^p−I_C⁰e^p) =τp

X

K∈T_ph

Z

K

r_K^p(e^p−I_C⁰e^p) +τp

X

K∈Tph

Z

K

(f(·, tp)−f_h^p)(e^p−I_C⁰e^p).

Using successively Cauchy-Schwarz’s inequality, the estimate (3.2) and the defini- tion 5.1 of the local estimator and the approximation term, we obtain

X

K∈Tph

Z

K

R^p_K(e^p−I_C⁰e^p). X

K∈Tph

hK(kr_K^pkK+kf(·, tp)−f_h^pkK)|e^p|_1,˜ω

K

. X

K∈Tph

(η^p_K+ξ_K^p)|e^p|_1,˜ω

K.

By discrete Cauchy-Schwarz’s inequality and the coerciveness ofa₂, we get X

K∈T_ph

Z

K

R^p_K(e^p−I_C⁰e^p).(η^p+ξ^p)|e^p|_1,Ω.(η^p+ξ^p)ke^pka₂. (5.3)

Similarly using (3.3) we estimate the edge residual term:

X

E∈E_ph^int

Z

E

J_E,n^p (e^p−I_C⁰e^p)6 X

E∈E_ph^int

J_E,n^p

_E

e^p−I_C⁰e^p _E

. X

E∈E_ph^int

J_E,n^p

_Eh^1/2_E |e^p|_1,˜ω

E

. X

K∈Tph

η_K^p |e^p|_1,˜ω

K. As before discrete Cauchy-Schwarz’s inequality yields

X

E∈E_ph^int

Z

E

J_E,n^p (e^p−I_C⁰e^p).η^p|e^p|_1,Ω.η^pke^pka₂. (5.4) Applying Cauchy-Schwarz’s and Young’s inequalities we get

a₁(e^p−1, e^p)6 1

2(ke^p−1k²_a

1+ke^pk²_a

1).

This estimate and (5.3), (5.4) in the identity (3.7) yield ke^pk²_a1+τpke^pk²_a2 6 1

2(ke^p−1k²_a1+ke^pk²_a1) +Cτp(η^p+ξ^p)ke^pka₂

6 1

2(ke^p−1k²_a

1+ke^pk²_a

1) +C²

2 τp(η^p+ξ^p)²+1

2τpke^pk²_a

2, for some constantC >0 depending only on the minimal angle ofT_ph, where in the second step we again use Young’s inequality. After simplification, this estimate is equivalent to

ke^pk²_a1+τpke^pk²_a2 6ke^p−1k²_a1+C²τp(η^p+ξ^p)²,

and we conclude by taking the sum on p= 1, . . . , n.

Corollary 5.3 (Second upper error bound). — The next estimate holds k∂t(uτ−uhτ)k²_L2(0,t_n;H₀¹(Ω).

n

X

p=1

τp((η^p)²+ (ξ^p)²) +ke⁰k²_a1. (5.5) Proof. — By the coerciveness ofa1, we have

k∂t(uτ−uhτ)(t)ka₁6 sup

v∈H₀¹(Ω)

a1(∂t(uτ−uhτ)(t), v)

kvk_a1 , for allt∈(t_p−1, tp). (5.6) Using the property

∂t(uτ−uhτ)(t) =e^p−e^p−1 τp

, for allt∈(t_p−1, tp), and the semi-discrete equation (2.8), for anyt∈(t_p−1, t_p) we may write

a1(∂t(uτ−uhτ)(t), v) =R^p(v)−a2(e^p, v), where the residualR^p is defined by

R^p(v) = (f(·, tp), v)−a1(u^p_h−u^p−1_h τp

, v)−a2(u^p_h, v), for allv∈H₀¹(Ω).