A posteriori error estimates for a nonconforming finite element discretization of the heat equation

Vol. 39, N 2, 2005, pp. 319–348 DOI: 10.1051/m2an:2005009

A POSTERIORI ERROR ESTIMATES FOR A NONCONFORMING FINITE ELEMENT DISCRETIZATION OF THE HEAT EQUATION

Serge Nicaise

and Nadir Soualem

Abstract. The paper presents ana posteriorierror estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation inR^d,d= 2 or 3, using backward Euler’s scheme.

For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results with minimal assumptions on the mesh. Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.

Mathematics Subject Classification. 65N15, 65N30, 65M50.

Received: July 12, 2004.

1. Introduction

This paper deals with the a posteriorianalysis of the heat equation approximated using backward Euler’s scheme in time and a (piecewise linear) nonconforming finite element approximation in space. There are several reasons to use nonconforming approximations. For example the approximation of the Stokes system requires the stability of the method, namely the discrete space has to satisfy the so-called inf-sup condition with a constant independent of the aspect ratio of the elements. Unfortunately standard conforming elements (like the mini element, the Taylor-Hood element, etc.) are not stable on anisotropic meshes (meshes for which the aspect ratio is no more bounded [3] and often used for the approximation of edge singularities and/or boundary layers), see [2, 4] and the references cited there. Therefore the use of nonconforming elements may be recommended since they are unconditionally stable [5].

As a first attempt we consider the case of the heat equation approximated by a piecewise linear nonconforming finite element space based on a regular family of triangulations. However our method may be extended to the Stokes system and to the use of anisotropic meshes. This will be investigated in forthcoming works.

In the conforming case several approaches have been introduced to define error estimators for the heat equation and the Stokes system [6–9, 17, 18, 20, 22]. To be able to extend these techniques to nonconforming spatial approximations, as for elliptic problems [14], we need to be able to estimate the consistency term appearing in the error equation. As in [14], this term is managed using a Helmholtz decomposition of the error.

This allows us to extend the results from [7–9, 22] to the nonconforming case.

Keywords and phrases. Error estimator, nonconforming FEM, heat equation.

1 Universit´e de Valenciennes et du Hainaut Cambr´esis, MACS, ISTV, 59313 Valenciennes Cedex 9, France.

Serge.Nicaise@univ-valenciennes.fr; Nadir.Soualem@univ-valenciennes.fr

c EDP Sciences, SMAI 2005

The schedule of the paper is the following one: Section 2 recalls the continuous and its discretizations. In Section 3 we give some analytical tools, in particular some properties satisfied by the spatial error and its Helmholtz like decomposition. Section 4 is devoted to the a posteriorianalysis of the time discretization. The efficiency and reliability of the spatial error estimator are established in Section 5. Thea posteriorianalysis of the full discrete problem is considered in Section 6, where we show the efficiency and reliability of the sum of the spatial and time error estimators. Finally in Section 7, we present some numerical tests which confirm our theoretical analysis. We further describe a space-time adaptive algorithm, which is validated by two relevant examples.

2. The continuous, time semi-discrete and full discrete problems

Let Ω be an open bounded ofR^d, d= 2 or 3, with a polygonal (d= 2) or polyhedral (d= 3) boundary Γ.

For the sake of simplicity, we assume that Ω is simply connected and that its boundary is connected. LetT be a positive and fixed real number.

Let us introduce some notation used in the whole paper: for shortness, ifDis a subset of Ω, theL²(D)-norm (resp. L²(D) inner product) will be denoted by · D (resp. (·,·)D). In the case D = Ω, we will drop the index Ω. The usual norm and seminorm of the standard Sobolev spaceH^s(D), withs >0, are denoted by·s,D

and| · |s,D.

In this paper we consider the following heat equation: Letube the solution of











∂u

∂t −∆u=f in Ω×]0, T[, u(., t) = 0 on Γ×]0, T[, u(.,0) =u0 in Ω.

(1)

The datum f is supposed to satisfy f ∈ L²(0, T;H⁻¹(Ω)) and the initial value u0 ∈ L²(Ω). Under these assumptions, problem (1) or equivalently

(∂tu(t), v) + (∇u(t),∇v) = (f(t), v),∀v∈H₀¹(Ω),∀ a.e. t∈(0, T), (2) has a unique (weak) solution inL²(0, T;H₀¹(Ω))∩C([0, T];L²(Ω)).

2.1. Time discretization using Euler’s scheme

We now suppose that f ∈ C([0, T];H⁻¹(Ω)). We further introduce a partition of [0, T] into subintervals [tp−1, tp], 1≤p≤N such that 0 =t0< t1 <· · · < tN =T. Denote byτp =tp−tp−1 the length of [tp−1, tp] and byτ= max_pτp the global time mesh size.

The semi-discrete approximation of the continuous problem (1) by a backward Euler scheme consists in finding a sequence (u^p)_0≤p≤N solution of













u^p−u^p−1

τp −∆u^p=f^p in Ω 1≤p≤N,

u^p= 0 on Γ 1≤p≤N,

u⁰=u0 in Ω,

(3)

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

Ωu^pv+τp

Ω∇u^p· ∇v=

Ωu^p−1v+τp

Ωf^pv ∀v∈H₀¹(Ω). (4) The unique solvability of the variational formulation (4) is then a direct consequence of the Lax-Milgram lemma.

2.2. Full discretization

Problem (4) is now discretized by a nonconforming finite element method. For that purpose, for all p= 0,1,· · ·, N, let us fix a conforming mesh Tph of Ω which form a regular family of triangulations in Ciarlet’s sense ([11], p. 124),i.e., there existsσ >0 such that

hK/ρK≤σ,∀K∈Tph,

where we recall that hK is the diameter ofK and ρK is the diameter of the largest ball included into K. All elements are triangles or tetrahedra and will be denoted byK. For allp, we denote byhp= max

K∈TphhK. The set of all edges/faces ofTph is denoted byEph. LetE_ph^int be the set of interior edges/faces ofTphandEK be the set of the edges/faces of the elementK. Finally for an edge/faceE ∈ EK∩ EL we denote byhE= ¹₂(^d|K|_|E| +^d|L|_|E|), its mean height.

Introduce the Crouzeix-Raviart nonconforming finite element space:

X_ph⁰ ={v∈L²(Ω) :v|K∈P₁,∀K∈Tph,

E

v|K =

E

v|L,∀E∈ E_K∩ E_L∩ E_ph^int, K, L∈Tph,

E

v|K = 0,∀E∈ EK∩Γ, K∈Tph}.

The full discrete approximation of problem (1) using Euler’s scheme and the Crouzeix-Raviart nonconforming finite element, is then given by: given an approximationu⁰_h∈X_0h⁰ of u0, findu^p_h∈X_ph⁰ , 1≤p≤N, such that:

Ωu^p_hvh+τp

K∈Tph

K

∇u^p_h∇vh=

Ωu^p−1_h vh+τp

Ωf^pvh (5)

for allvh∈X_ph⁰ .

Note that the Crouzeix-Raviart elements were recently used in [1] for the discretization of a mixed formulation of the Laplace equation and that the nonconformity of the approximation also renders theira posteriorianalysis more delicate.

Definition 2.1. Letu^p be a solution of (4) andu^p_h a solution of (5), then we denote the spatial error by e^p=u^p−u^p_h.

Let us finish this section by introducing some useful notation and properties used below.

The notationab anda∼bmeans the existence of positive constants C1 and C2 (which are independent of the mesh size of the triangulations, of the time step size and of the function under consideration) such that a≤C2b andC1b≤a≤C2b, respectively.

For a boundary edge/faceEwe denote the outward normal vector bynE. In 2D, we further define the tangent vector bytE = (−nE2, nE1) ifnE= (nE1, nE2). Given an interior edge/faceE, we choose an arbitrary normal directionnE and denote byKinandKextthe two elements sharing this edge/face. Without any restriction, we may suppose here that nE is pointing to Kext like in Figure 1. In 2D, denote as before the tangent vector by tE= (−n_E2, nE1) ifnE = (nE1, nE2).

For the analysis of the nonconforming approximation, we will use the following Crouzeix-Raviart property:

E

uh

E= 0 ∀E∈ Eph,∀uh∈X_ph⁰ , (6)

@@

@@XXXXXXXXXXXXXXX

n

K

K

Figure 1. Two elements sharing the edgeE. where the jump of some functionvacross 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∈ Eph\ E_ph^int. Note that the sign of

v(x)

E depends on the orientation of nE. However, quantity like a gradient jump ∇v·nE

E is independent of this orientation.

For a functionv∈X_ph⁰ we define its broken gradient∇hv by (∇hv)_|K =∇(v|K),∀K∈Tph.

In the sequel we will use local patches: for an elementK we defineωK as the union of all elements having a common edge/face withK, for an edge/faceE, letωE be the union of both elements havingEas edge/face and finally for a nodex, letωxbe the union of both elements havingxas node. Similarly denote by ˜ωK (resp. ˜ωE) the union of all triangles sharing a node withK (resp. E).

We further need the standardP₁ conforming finite element spaces Vph ={v∈H¹(Ω) :v|K ∈P₁,∀K∈Tph}, V_ph⁰ =Vph∩H₀¹(Ω).

For our error analysis we require an interpolant that mapsX_ph⁰ ⊕H₀¹(Ω) toV_ph⁰. Hence Lagrange interpolation operator is unsuitable but Cl´ement like interpolation operator is more appropriate. To write the results in the largest setting as possible, let us set

Yph ={v∈L²(Ω) :v|K∈H¹(K),∀K∈Tph,

Ev|K =

Ev|L,∀E∈ EK∩ EL∩ E_ph^int, K, L∈Tph,}, Y_ph⁰ ={v∈L²(Ω) :v|K∈H¹(K),∀K∈Tph,

E

v|K =

E

v|L,∀E∈ EK∩ EL∩ E_ph^int, K, L∈Tph,

E

v|K = 0,∀E∈ E_K∩Γ, K∈Tph}.

Note thatH¹(Ω)⊂Yph and thatX_ph⁰ ⊕H₀¹(Ω)⊂Y_ph⁰.

Recall that the Cl´ement interpolation operator is defined as follows: Denote by Nph the set of nodes of the triangulationTphand byN_ph^int the set of interior nodes of the triangulationTph. For each nodexdenote further byλxthe standard hat function associated with x, namelyλx∈Vph and satisfies

λx(y) =δx,y,∀y∈ Nph.

For anyv∈Yph andw∈Y_ph⁰, we defineICv andI_C⁰wby ICv =

x∈Nph

|ωx|⁻¹

ωx

v

λx, (7)

IC⁰w=

x∈N_ph^int

|ωx|⁻¹

ωx

w

λx. (8)

Note thatICvbelongs toVph, whileI_C⁰wbelongs toV_ph⁰. Moreover these operators have the following properties:

Lemma 2.2. For all v∈Yph andw∈Y_ph⁰, we have

v−ICv_K hK∇hvω_˜K,∀K∈Tph, (9) v−ICvE h^1/2_E ∇hvω˜E,∀E∈ Eph, (10) w−IC⁰w

K hK∇hwω˜K,∀K∈Tph, (11)

w−I_C⁰wE h^1/2_E ∇hwω_˜E,∀E∈ E_ph^int, (12)

∇I_C⁰wK ∇hwω˜K,∀K∈Tph. (13) Proof. Forv and w in H¹(Ω), the above properties are proved in [12] (see also [19, 21] for other interpolation operators) using scaling arguments, but a careful analysis of their proof reveals that these properties hold forv

andwas in the statement of the Lemma.

The mean value of some functionv on an edge/faceE is defined by ME(v) = 1

|E|

E

v.

In the sequel we often need the following Green’s formulas: if D is a bounded open subset of R² and v, w ∈ H¹(D), then we have

D

∇v·curl w=

∂D

vcurl w·n=

∂D

∇v·tw, (14)

wheretis the unit tangent vector along∂Dandcurl wis the vectorial curl ofw, namelycurl w=

∂2w

−∂1w

. Similarly ifD is a bounded open subset ofR³ andv∈H¹(D),w∈H¹(D)³ then we have

D

∇v·curl w=

∂D

vcurl w·n=

∂D

(∇v×n)·w. (15)

We finally introduce the gradient jump of u^p_h in normal and tangential direction by J_E,n^p =

∇u^p_h·nE

E ifE∈ E_ph^int, 0 ifE∈ Eph\ E_ph^int. Ifd= 2, then

J_E,t^p =

∇u^p_h·tE

E ifE∈ E_ph^int,

−∇u^p_h·tE ifE∈ Eph\ E_ph^int. Ifd= 3, then

J_E,t^p =

∇u^p_h×nE

E ifE∈ E_ph^int,

−∇u^p_h×nE ifE∈ Eph\ E_ph^int.

3. Some analytical tools

In this section we collect different properties satisfied by the spatial errore^p that we will use in the proof of the spatial error bounds.

Lemma 3.1 (Galerkin orthogonality). The error e^p satisfies the Galerkin orthogonality relation

K∈Tph

K

∇he^p· ∇vh=

Ω

e^p−1−e^p

τp vh,∀vh∈V_ph⁰. (16) Proof. It suffices to subtract (4) withv=vh∈V_ph⁰ to the identity (5).

Lemma 3.2. Let ϕ∈H¹(Ω)if d= 2andϕ∈H¹(Ω)³ if d= 3. Then the error satisfies the following identity

Ω∇he^p·curl ϕ=

E∈Eph

E

J_E,t^p ·ϕ. (17)

Proof. Assume thatd= 2. Integrations by parts in Ω and in each elementK give (see (14))

Ω∇he^p·curl ϕ=

Ω∇u^p·curl ϕ−

K∈Tph

K

∇u^p_h·curl ϕ

=

Γcurl ϕ·nu^p−

K∈Tph

∂K

∇u^p_h·tKϕ.

Asu^p∈H₀¹(Ω) andϕ∈H¹(Ω), we conclude using the definition ofJ_E,t^p .

The proof is similar in dimension 3 using (15).

Lemma 3.3 (error orthogonality). The error satisfies

K∈Tph

K

∇he^p·curl ϕh= 0,∀ϕh∈Vph if d= 2andϕh∈(Vph)³ ifd= 3. (18)

Proof. Consider an arbitrary elementϕh inVphifd= 2 or in (Vph)³ifd= 3. As before, by integrating by parts (cf. the identities (14) and (15)), we obtain (recalling thatu^p∈H₀¹(Ω))

K∈Tph

K

∇he^p·curl ϕh =

Ω∇u^p·curl ϕh−

K∈Tph

K

∇hu^p_h·curl ϕh

=

Γu^pcurl ϕh·n−

K∈Tph

∂K

u^p_hcurl ϕh·nK

=−

K∈Tph

∂K

u^p_hcurl ϕh·nK

=−

E∈Eph

E

u^p_h

Ecurl ϕh·nE

=−

E∈Eph

(curl ϕh·nE)

E

u^p_h

E,

since the function (curl ϕh·nE)|E is constant on E ∈ Eph. The Crouzeix-Raviart property (6) satisfied by

u^p_h∈X_ph⁰ allows us to finish the proof.

Lemma 3.4. The error e^p satisfies

K∈Tph

K

∇he^p· ∇w=

K∈Tph

K

f^p−u^p−u^p−1 τp

w+

E∈Eph

E

J_E,n^p w,

for any w∈H₀¹(Ω).

Proof. Elementwise integration by parts and the observation that ∆u^p_h= 0 on allK∈Tph yield

K∈Tph

K

∇he^p∇w=

Ω∇u^p· ∇w−

K∈Tph

K

∇hu^p_h· ∇w

=

Ω

f^p−u^p−u^p−1 τp

w

−

K∈Tph

−

K

∆u^p_hw+

∂K

n· ∇u^p_hw

=

Ω

f^p−u^p−u^p−1 τp

w

−

K∈Tph

E∈EK

E

n· ∇u^p_hw.

We conclude by using the definition ofJ_E,n^p and the continuity ofwthrough the edges/faces.

Corollary 3.5. For any w∈H₀¹(Ω) andϕ∈H¹(Ω)if d= 2andϕ∈H¹(Ω)³ if d= 3, we have

K∈Tph

K

∇he^p·(∇w+curl ϕ) =

K∈Tph

K

f^p−u^p−u^p−1 τp

w+

E∈Eph

E

(J_E,n^p w+J_E,t^p ·ϕ). (19)

Proof. Direct consequence of Lemmas 3.2 and 3.4.

We now recall the following result (see Lem. 3.2 of [14] in 2D and [13] in 3D or [15], Chap. I):

Lemma 3.6 (Helmholtz decomposition of the error). We have the following error decomposition

∇_he^p=∇w^p+curl ϕ^p, (20) with ϕ^p∈H¹(Ω) ifd= 2 andϕ^p∈(H¹(Ω))³ if d= 3andw^p∈H₀¹(Ω). Moreover the next estimates hold:

|w^p|_1,Ω≤ ∇he^p, (21)

|ϕ^p|_1,Ω∇he^p. (22) Proof. We consider the following Dirichlet problem: findw^p∈H₀¹(Ω) solution of

div (∇he^p− ∇w^p) = 0 in Ω,

w^p = 0 on Γ. (23)

The weak formulation of that problem (23) is:

Ω∇w^p· ∇v=

Ω∇he^p· ∇v, ∀v∈H₀¹(Ω). (24)

As the vector field∇he^p− ∇w^p is divergence free in Ω,i.e.,

div (∇he^p− ∇w^p) = 0 in Ω.

By Theorem I.3.1 of [15] if d = 2 or Theorem I.3.4 of [15] if d = 3, there exists ϕ^p ∈ H¹(Ω) if d = 2 and ϕ^p∈(H¹(Ω))³ ifd= 3 such that

curl ϕ^p=∇he^p− ∇w^p.

The estimate (21) directly follows by using (24) withv=w^p. The second estimate (22) is obtained as follows.

Using the expansion (20), we may write

Ω|curl ϕ^p|² =

Ωcurl ϕ^p·curl ϕ^p

=

Ωcurl ϕ^p·(∇he^p− ∇w^p). By Green’s formula and the boundary conditionsw^p= 0 on Γ, we obtain

Ω|curl ϕ^p|²=

Ωcurl ϕ^p· ∇he^p. (25)

By Cauchy-Schwarz’s inequality we conclude

curl ϕ^p ≤ ∇he^p. (26) Ifd= 2 the estimate (22) directly follows from the above estimate since|ϕ^p|_1,Ω=curl ϕ^p. Ifd= 3, we may notice that the application of the closed graph theorem yields a vector field ϕ^p satisfying

ϕ^p_1,Ωcurl ϕ^p. (27)

Indeed it suffices to consider the mapping

F :H¹(Ω)³/K→ {w∈L²(Ω)³:div w= 0}:ϕ→curl ϕ,

where K={ϕ∈H¹(Ω)³ :curl ϕ= 0}. This mapping is continuous and bijective (by Thm. I.3.4 of [15]) and consequently by the closed graph theorem, its inverse mapping is also continuous.

Therefore we may conclude as before using the above estimates (26) and (27).

The above lemmas allow us to prove the

Lemma 3.7. The following identities hold

τp

Ω∇he^p· ∇w^p=τp

Ω

f^p−u^p_h−u^p−1_h τp

(w^p−IC⁰w^p) +τp

E∈Eph

EJ_E,n^p (w^p−IC⁰w^p)−

Ω(e^p−e^p−1)w^p, (28)

Ω∇he^p·curl ϕ^p=

E∈Eph

E

J_E,t^p ·(ϕ^p−ICϕ^p), (29)

e^p2+τp

Ω|∇he^p|²= (e^p−1, e^p) + (e^p−e^p−1, e^p−w^p−I_C⁰(e^p−w^p)) (30) +τp

Ω∇he^p· ∇I_C⁰(e^p−w^p) +τp

Ω

f^p−u^p_h−u^p−1_h τp

(w^p−IC⁰w^p) +τp

E∈Eph

E

(J_E,n^p (w^p−IC⁰w^p) +J_E,t^p ·(ϕ^p−ICϕ^p)).

Proof. The identity (28) follows from the Galerkin relation of Lemma 3.1 withvh=I_C⁰w^p∈V_ph⁰ and Lemma 3.4.

The second identity (29) is a consequence of the orthogonality relation of Lemma 3.3 with ϕh = ICϕ^p and Lemma 3.2.

Using the error decomposition (20) we may write τp

Ω|∇he^p|²=τp

Ω∇he^p·(∇w^p+curl ϕ^p). Therefore the identities (28), (29) directly lead to

τp

Ω|∇he^p|²=−

Ω(e^p−e^p−1)w^p+τp

Ω

f^p−u^p_h−u^p−1_h τp

(w^p−I_C⁰w^p) +τp

E∈Eph

E

(J_E,n^p (w^p−I_C⁰w^p) +J_E,t^p ·(ϕ^p−ICϕ^p)).

This identity may be equivalently written e^p2+τp

Ω|∇he^p|² = (e^p−1, e^p) + (e^p−e^p−1, e^p−w^p−I_C⁰(e^p−w^p)) + (e^p−e^p−1, I_C⁰(e^p−w^p)) +τp

Ω

f^p−u^p_h−u^p−1_h τp

(w^p−IC⁰w^p) +τp

E∈Eph

E

(J_E,n^p (w^p−IC⁰w^p) +J_E,t^p ·(ϕ^p−ICϕ^p)).

This identity and the Galerkin orthogonality relation (16) lead to (30).

4. A

analysis of the time discretization

Inspired from [7, 9, 16, 17], we define the time error indicators:

η^p_t =τ_p^1/2∇h(u^p_h−u^p−1_h ),1≤p≤N. (31)

The only difference with the above papers relies on the fact thatu^p_h−u^p−1_h is not inH¹(Ω). Since the continuous problems (2) and (4) are not related to the approximation spacesX_ph⁰ , we easily adapt the arguments used there to our setting.

Note that in (31) we have written∇h(u^p_h−u^p−1_h ) whileu^p_handu^p−1_h are not related to the same triangulation, butu^p_h−u^p−1_h may be seen as a piecewiseP1-function on the “triangulation”Tph∩Tp−1,hmade of the intersections of elements from Tph with elements fromTp−1,h. The broken gradient is then calculated on this triangulation Tph∩Tp−1,h.

For shortness we introduce the following notation: Denote by πτf the step function which is constant and equal tof(tp) on each interval (tp−1, tp), 1≤p≤N. For a sequencev^p∈X_ph⁰ ⊕H₀¹(Ω), 0≤p≤N, we denote byvτ its “Lagrange” interpolant, which is affine on each interval [tp−1, tp], 1 ≤p≤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,∀t∈[tp−1, tp],1≤p≤N.

Denote finallyeτ=u−uτ, the time discretization error.

As

∂tuτ= u^p−u^p−1 τp

on (tp−1, tp), the semi-discrete equation (4) is equivalent to

(∂tuτ(t), v) + (∇u^p,∇v) = (f^p, v),∀v ∈H₀¹(Ω),∀t∈(tp−1, tp). (32) Taking the difference with (2), we derive the residual equation

(∂teτ(t), v) + (∇eτ(t),∇v) = ((f−f^p)(t), v) + (∇(u^p−uτ)(t),∇v),∀v ∈H₀¹(Ω),∀a.e. t∈(tp−1, tp). (33) This equation allows to prove the

Theorem 4.1 (time upper error bound). The next estimate holds eτ(tn)²+

_tn

0 ∇eτ(s)²ds n p=1

(η^p_t)²+ _tn

0 ∇h(uτ−uhτ)(s)²ds+f−πτf²_L2(0,tn;H⁻¹(Ω)). (34) Proof. The residual equation (33) yields (see Prop. 3.1 of [9])

eτ(tn)² + _tn

0 ∇eτ(t)²dt≤2 n p=1

_tp

tp−1

∇(u^p−uτ)(s)²ds+ 2f−πτf²_L2(0,tn;H⁻¹(Ω)). (35)

By the definition ofuτ we clearly have _tp

tp−1

∇(u^p−uτ)(s)²ds=τp

3 ∇(u^p−u^p−1)². (36)

Using the triangular inequality, we simply write

τ_p^1/2∇(u^p−u^p−1) ≤η^p_t +τ_p^1/2∇h(u^p−u^p_h)+τ_p^1/2∇h(u^p−1_h −u^p−1_h ).

Moreover the arguments from Lemma 2.3 of [9] yields

τp∇h(u^p−u^p_h)²+τp∇h(u^p−1_h −u^p−1_h )² _tp

tp−1

∇h(uτ−uhτ)(s)²ds. (37)

The above identity and these two estimates yield _tp

tp−1

∇(u^p−uτ)(s)²ds(η_t^p)²+ _tp

tp−1

∇h(uτ−uhτ)(s)²ds. (38)

This estimate in (35) leads to the conclusion.

Corollary 4.2 (second time upper error bound). The next estimate holds

∂teτ²_L2(0,tn;H⁻¹(Ω))ⁿ

p=1

(η^p_t)²+ _tn

0 ∇h(uτ−uhτ)(s)²ds+f−πτf²_L2(0,tn;H⁻¹(Ω)). (39) Proof. The residual equation (33) directly gives

∂teτ²_L2(0,tn;H⁻¹(Ω))f−πτf²_L2(0,tn;H⁻¹(Ω))+ _tn

0 ∇eτ(s)²ds+ n p=1

_tp

tp−1

∇(u^p−uτ)(s)²ds.

The second term of this right-hand side is estimated in (34), while the third term is estimatedvia(38).

The local time upper bound is even easier to prove:

Theorem 4.3 (time lower error bound). For allp= 1,· · · , N, the next estimate holds η^p_t ∇heτ_L²_(tp−1,tp;L²_(Ω))+∂teτ_L²_(tp−1,tp;H⁻¹_(Ω))

+τ_p^1/2(∇h(u^p−u^p_h)+∇h(u^p−1−u^p−1_h )) +f−πτf_L²_(tp−1,tp;H⁻¹_(Ω)). (40) Proof. By the triangular inequality we may write

η^p_t τ_p^1/2(∇(u^p−u^p−1)+∇_h(u^p−u^p_h)+∇_h(u^p−1−u^p−1_h )).

The estimation of the term τp^1/2∇(u^p−u^p−1) is made as in Proposition 3.3 of [9] by using the identity (36) (withn=p) and takingv=u^p−uτ in (33) and integrating the result int∈(tp−1, tp).

5. A

analysis of the spatial discretization 5.1. Upper error bound

The exact element residual is given by

f^p−u^p_h−u^p−1_h τp

· As usual [20] it is replaced by an approximate element residual

f_h^p−u^p_h−u^p−1_h τp ,

wheref_h^p is a finite dimensional approximation off^p (for instance (f_h^p)_|K := _|K|¹

Kf^p, for allK∈Tph).

Definition 5.1. Letp≥1. The local error estimatorη^p_K is defined by η^p_K =hK

f_h^p−u^p_h−u^p−1_h τp

K

+

E∈EK

h^1/2_E J_E,n^p

E+J_E,t^p

E

.

The global spatial error estimatorη^p is given by

(η^p)²=

K∈Tph

(η^p_K)².

The local and global approximation terms are defined by

ξ_K^p =hKf^p−f_h^p_ωK, (ξ^p)²=

K∈Tph

(ξ^p_K)².

Theorem 5.2 (upper error bound). The next estimate holds eⁿ²+

n p=1

τp∇he^p2 n p=1

max{h²_p, τp}(η^p)²+e⁰²+ n p=1

τp(ξ^p)². (41)

Proof. This upper bound is a consequence of Lemmas 3.6 and 3.7. We first estimate some terms of the right- hand side of the identity (30) of Lemma 3.7. Using successively Cauchy-Schwarz’s inequality, the estimate (11) and the definition 5.1 of the local estimator, we obtain

K∈Tph

K

f_h^p−u^p_h−u^p−1_h τp

(w^p−I_C⁰w^p)

K∈Tph

hK

f_h^p−u^p_h−u^p−1_h τp

K

|w^p|_1,˜ωK

K∈Tph

η^p_K|w^p|_1,˜ωK.

By discrete Cauchy-Schwarz’s inequality we get

K∈Tph

K

f_h^p−u^p_h−u^p−1_h τp

(w^p−I_C⁰w^p)η^p|w^p|_1,Ω. (42)

Similarly using (12) and (10) we estimate the term with the jumps of normal and tangential derivatives:

E∈Eph

E

J_E,n^p (w^p−I_C⁰w^p)≤

E∈Eph

J_E,n^p

E

w^p−I_C⁰w^p

E

E∈Eph

J_E,n^p

Eh^1/2_E |w^p|_1,˜ωE

K∈Tph

η^p_K|w^p|_1,˜ωK,

E∈Eph

E

J_E,t^p ·(ϕ^p−ICϕ^p)≤

E∈Eph

J_E,t^p

Eϕ^p−ICϕ^p_E

E∈Eph

J_E,t^p

Eh^1/2_E |ϕ^p|_1,˜ωE

K∈Tph

η^p_K|ϕ^p|_1,˜ωK.