Vol. 40, No 6, 2006, pp. 991–1021 www.edpsciences.org/m2an
DOI: 10.1051/m2an:2006034
ROBUST A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT DISCRETIZATIONS OF THE HEAT EQUATION WITH DISCONTINUOUS
COEFFICIENTS∗
Stefano Berrone
1Abstract. In this work we derivea posteriorierror estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stableθ-scheme with 1/2≤θ≤1.
Following remarks of [Picasso,Comput. Methods Appl. Mech. Engrg. 167(1998) 223–237; Verf¨urth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space- discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicitycondition [Dryja et al., Numer. Math.
72(1996) 313–348; Petzoldt,Adv. Comput. Math. 16(2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.
Mathematics Subject Classification. 65M60, 65M15, 65M50.
Received: July 15, 2005. Revised: October 24, 2006.
1. Introduction
In many practical applications a heat conduction problem involving a non homogeneous medium, made for example by different materials, has to be solved. To deal with these situations, we propose robusta posteriori error estimates for the heat equation with discontinuous, piecewise constant coefficients based on a discretization by conforming finite elements and the classical A-stable θ-scheme with 1/2≤θ≤1.
Since the pioneering work of Babuˇska and Rheinboldt [1]a posteriorierror estimates and adaptive algorithms have become an important field for scientific computing [2, 7, 10, 12, 16, 18] and many works were devoted to parabolic problems [3, 9, 14, 17].
The estimator here derived is based on equation residuals as in [3, 14, 17]. In [14] residual based error estimators bound the error measured in the normt[n]
t[n−1] ∇.20dtfrom above and from below; the implicit Euler
scheme with linear finite elements is considered and only refinement is allowed; further, a condition between the meshsize and the timestep-length has to be satisfied. In [17] residual based error estimators bounding the error containing also the termt[n]
t[n−1]∂ .
∂t 2
−1dtare presented for constant diffusivity coefficients. The proof of
Keywords and phrases. A posteriorierror estimates, parabolic problems, discontinuous coefficients.
∗This work was supported by Italian fundsMiur-PRIN-2004 “Adattivit`a e avanzamento in tempo nei modelli numerici alle derivate parziali” andIndam-GNCS-2005 “Metodi numerici per lo studio di problemi evolutivi multiscala”.
1 Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
sberrone@calvino.polito.it
c EDP Sciences, SMAI 2007
Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006034
the upper bound is performed deriving an upper bound of the error estimator in terms of a “time functional residual” and a “space functional residual”. In [3] linear and semilinear heat equations are considered: first, a semidiscretization in time by implicit Euler scheme is introduced; second, a discretization in space by finite elements is performed. This approach is considered in order to uncouple as much as possible the time and the space errors. Continuously differentiable diffusivity coefficients are considered.
In our work we consider the case of discontinuous coefficients. We use a full discretization approach instead of considering a semidiscrete formulation as in [3]. Our estimates allow us to perform a control of the space- discretization used in each time slab and of the timestep-length. The ratio between the upper and the lower bounds for the error is independent on any meshsize, timestep, diffusivity parameter and its jumps across the domain. To ensure this robustness with respect to the parameters jumps thequasi-monotonicitycondition [8,13]
is assumed. In the proof of the lower bound we introduce an orthogonal space of edge bubble functions. In our estimates we consider the data-approximation error and we propose a splitting of this error in two terms:
a data-approximation error in space and a data-approximation error in time; this splitting can be used in the adaptation of the mesh and in the choice of the timestep-length in each time slab.
In Section 4 we present some numerical results on uniform meshes and constant timestep-lengths to carefully analyze the behaviour of the effectivity indices and prove robustness of the estimates. These results also confirm that the terms forming the error estimator and the data-approximation error mainly depend either on the space discretization or on the time discretization. This splitting can be very useful in an adaptive algorithm to adapt mesh and timestep-length. A simple adaptive strategy and some preliminary numerical results are proposed in the Appendix.
2. The heat equation
2.1. The continuous problemLet Ω be a polygonal domain inR2with boundary∂Ω and let (0,Ξ) be the time interval of interest. For any f∈L2(0,Ξ; L2(Ω) ) andu[0]∈L2(Ω), we want to findu: Ω×(0,Ξ)→Rsuch that
∂u
∂t −∇ ·(κ∇u) = f, in Ω×(0,Ξ), (1)
u(x, t) = 0, on∂Ω×(0,Ξ), (2)
u(x,0) = u[0](x), in Ω. (3)
The diffusivity parameter κ(x), 0 < κmin ≤ κ ≤ κMax < ∞, is a function constant in time and piecewise constant on the polygonal subdomains Ωd,d= 1, . . . , D, with∪Dd=1Ωd= Ω and Ωi∩Ωj=∅,∀i=j.
Setting W =
w∈L2(0,Ξ; H10(Ω) ) : ∂w∂t ∈L2(0,Ξ; H−1(Ω) )
the variational formulation of the above prob- lem is: F ind u∈W such that u(.,0) =u[0] and
∂u
∂t, v
+ (κ∇u,∇v) = (f, v), ∀v ∈H10(Ω), a.e.in (0,Ξ). (4) Here. , .stands for the duality pairing between H−1(Ω) and H10(Ω),(. , .) is the usual inner product in L2(Ω).
Ifu∈W, thenu∈C0([0,Ξ]; L2(Ω) ) and the initial condition u(.,0) =u[0]is meaningful in L2(Ω).
2.2. The numerical discretization
Let us consider a partition of (0,Ξ) into subintervals
t[n−1], t[n]
of length ∆t[n] = t[n]−t[n−1], with 0 = t[0] < t[1] < · · · < t[N] = Ξ; set I[n] = t[n−1], t[n]
. In each time-slab Ω×I[n], n ≥ 1, we consider a regular family of partitions Th[n] of Ω into trianglesT∈ Th[n] which satisfy the usual conformity and minimal- angle conditions [5], we denote by h[n]T the diameter of each elementT∈Th[n] and byh[n] the maximum ofh[n]T
over all the elements T ∈ Th[n]. From now on the subscript hstands for h[n]. We assume that each triangu- lation Th[n] induces triangulations Th,d[n] of the subdomains Ωd, d = 1, .., D, such that Th[n] = ∪Dd=1Th,d[n]. Let Vh[n] =
vh∈H10(Ω)∩ C0(Ω) : v|T∈Pk(T), ∀T∈Th[n]
⊂V = H10(Ω) be a family of conforming finite element spaces based on the partitions Th[n]. We denote by Pk(T) the space of polynomials of degree k ≥ 1 on the element T∈ Th[n]. Assuming u[0]∈L2(Ω), we define u[0]h,∆t=Ph,k[1]u[0] to be the L2(Ω)-projection ofu[0] on the finite element space V[1]h defined onTh[1]. The subscripth,∆tis used to refer to the full discretization in space and in time. Then, we introduce the discretization based on the classical θ-scheme for the time integration:
F ind u[n]h,∆t∈V[n]h such that∀vh∈V[n]h , n= 1, ..., N u[n]h,∆t−u[n−1]h,∆t
t[n]−t[n−1] , vh
+θ
κ∇u[n]h,∆t,∇vh
+ (1−θ)
κ∇u[n−1]h,∆t ,∇vh
=θ
ΠTf[n], vh
+ (1−θ)
ΠTf[n−1], vh
, ∀vh∈Vh[n]. (5)
In the last scalar products of the previous equation we assume that f∈ C0([0,Ξ]; L2(Ω) ) and we set f[r] = f(., t[r]), r∈ {n−1, n}. Moreover we introduce an arbitrary piecewise polynomial approximation ΠTf of the data f. If the initial conditionu[0] belongs toC0(Ω), instead of the projection operatorPh,k[1], we can use the interpolation operatorπ[1]h,k:C0(Ω)→Vh[1].
At last we define the continuous, piecewise affine in time approximation of the solutionu(., t):
uh,∆t(x, t) = t−t[n−1]
t[n]−t[n−1] u[n]h,∆t(x) + t[n]−t
t[n]−t[n−1]u[n−1]h,∆t (x), x∈Ω, t∈I[n], n= 1, ..., N. (6)
3. A residual-based
A POSTERIORIerror estimator
In this section we derive a residual-based error estimator for our fully discretized model problem following the work in [14, 17]. In particular, we shall derive a global-in-space local-in-time upper and lower bounds. At first, we introduce some notation which will be used for the construction of the estimator.
3.1. Definitions and general results
For each time-slab Ω×I[n] we define a partitionsTh[n−1,n] that is a common refinement ofTh[n−1] andTh[n], satisfying conformity and minimal angle condition and atransition conditionor moderate coarsening condition [17]: there exists a constantsCtr such that
sup
n=1,...,N
sup
T∈Th[n]
sup
T∗∈Th[n−1,n]:T∗⊆T
h[n]T h[n−1,n]T∗
≤Ctr, (7)
being h[n−1,n]T∗ the diameter of the element T∗ ∈ Th[n−1,n]. For anyT∈ Th[n−1,n] we denote byE(T) the set of its edges; we denote byEh[n−1,n] =∪T∈T[n−1,n]
h E(T) the set of all edges of the triangulationTh[n−1,n]. Moreover, we split Eh[n−1,n] into the form Eh[n−1,n] = Eh,Ω[n−1,n]∪ Eh,∂Ω[n−1,n] with Eh,Ω[n−1,n] =
E∈Eh[n−1,n]:E ⊂∂Ω , and Eh,∂Ω[n−1,n] =
E∈Eh[n−1,n]:E⊂∂Ω
. Similarly, we define the corresponding setsEh[n],Eh,Ω[n] andEh,∂Ω[n] of edgesE ofTh[n]. For any edgeE∈Eh[n−1,n] and we define:
ω[n]E =
{T∈Th[n−1,n]:E∈E(T)}
T.
x2
x1 ac
T1 T0
FE,T a0
a1
a2
a1 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000
11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111
x x2
T
T
a0 a1 1
a2
E
E
T2
Figure 1. The mappingFE,T : ˆT →T2.
T
T ac
ac
00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000
11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000
111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111 111111111
x x
1 2
a0
a1
a2
a2
a0
a1
T T
2 2
E
Figure 2. The support of the func- tionb[n]E.
To any edgeE ∈ Eh,Ω[n−1,n] we associate an orthogonal unit vectornE and denote by [[.]]E the jump acrossE in the directionnE. Let us denote by ˆT the reference triangle and by ˆEthe reference edge as shown in Figure 1 on the left. Let λi,i= 0,1,2 be the barycentric coordinates on the reference triangle, then thereference triangle bubble functionis ˆbTˆ= 27λ0λ1λ2, and thereference edge bubble functionis ˆbEˆ= 4ˆx(1−ˆx−y) andˆ FT[n]: ˆT →T is the affine mapping from the reference triangle to the triangleT ∈ Th[n−1,n] [5, 15]. For sake of simplicity, we will drop the superscript [n] in the symbols of the mappings. For anyT ∈ Th[n−1,n] we indicate withb[n]T the triangle bubble functiondefined byb[n]T = ˆbTˆ◦FT−1. Note that this bubble function does not depend on time in each time-slab.
Given anyE∈Eh,Ω[n−1,n], letTandTthe two triangles ofTh[n−1,n]such thatωE[n]=T∪T. Let us enumerate the vertices ofT and T counterclockwise in such a way that the vertices ofE are numbered first. LetT be one of the trianglesTandT, assume thatEhas verticesa0anda1and denote byac= (xc, yc) the barycentre of the triangleT; let us partitionT into the trianglesT0,T1,T2 with T2 havingE as a side (see Fig. 1). Let FE,T : ˆT →T2 be the invertible affine mapping that maps the reference triangle ˆT onto the triangleT2
FE,T(ˆx,y) =ˆ a0λ0(ˆx,y) +ˆ a1λ1(ˆx,y) +ˆ acλ2(ˆx,y)ˆ , if (ˆx,y)ˆ ∈T .ˆ Then we define theedge bubble functionb[n]E by patching the two bubble functions:
b[n]E,T = ˆbEˆ◦FE,T−1, b[n]E,T = ˆbEˆ◦FE,T−1,
each one being nonzero only on T2 and T2, respectively. Finally, let us define the set ω[n]E =T2∪T2 (dashed area in Fig. 2). For the boundary edge E that belongs to the elementT only, we naturally identifyb[n]E with b[n]E,T = ˆbEˆ◦FE,T−1.
With this definition of edge bubble functions we have a set of orthogonal functions, in the sense that the intersection of the supports of two different edge bubble functions is the empty set or a whole segment. This property is also true for the set oftriangle bubble functions.
Moreover, for the reference edge ˆE we define the extension operator ˆPEˆ : Pi( ˆE) →Pi( ˆT) which extends a polynomial of degree idefined on the edge ˆE to a polynomial of the same degree defined on ˆT with constant values along lines orthogonal to the edge ˆE. Then, we define the extension operator PE : Pi(E) →Pi(ω[n]E ) which extends a polynomial of degree i defined on the edge E to a piecewise polynomial of the same degree
defined onω[n]E by patching the two operators:
PE|
T =FE,T
2 ◦PˆEˆ◦F−1
E,T2|E, PE|T =FE,T
2 ◦PˆEˆ◦F−1
E,T2|E.
The extension operator PE is continuous, but not C1 in ω[n]E. In the following we will need to collect all the triangles belonging to some set ω[n]E , so let us define T[n−1,n]
h,ω =
T2∈ω[n]E :E∈ Eh[n−1,n]
.
The symbol ab means that there exists a constantc independent of any meshsize, timestep, parameter and jump of parameters such thata≤c b. The symbolabmeans that ab andba.
For any time interval I[n], T ∈ Th[n−1,n] and E ∈ Eh[n−1,n], the bubble functions b[n]T and b[n]E have the following properties: supp b[n]T = T, 0 ≤ b[n]T ≤1, max(x,y)∈Tb[n]T(x, y) = 1; supp b[n]E = ω[n]E , 0 ≤ b[n]E ≤1, max(x,y)∈Eb[n]E(x, y) = 1; b[n]T 2
0,T |T|, b[n]E 2
0,T |T|, b[n]E 2
0,E |E|. Thanks to the regularity hypothesis on Th[n−1,n], there exist constants depending on the smallest angle in the triangulation, but not on the mesh size, such that for each n = 1, ..., N we have: |T|
h[n−1,n]T 2
, ∀T ∈ Th[n−1,n], h[n−1,n]T h[n−1,n]E , ∀E∈ E(T) and|T|
h[n−1,n]E 2
, ∀T ∈ω[n]E .
In the following κT will denote the constant value of κ in the triangle T ∈ Th[n], ˆκω[n]
E is the maximum of the values of κT over the two triangles T ∈ Th[n] sharing the edge E (we will use the same symbol to denote the maximum ofκT over the two trianglesT ∈ Th[n−1,n] sharing the edgeE ∈ Eh[n−1,n], it will be clear from the context which situation we are referring to). Moreover we shall use a modified quasi-interpolation operator Ih : V → V[n]h like the quasi-interpolation operator of Cl´ement, [6]. The definition of this kind of interpolation operator requires the quasi-monotonicityhypothesis [8, 13] ofκ(x) with respect to any nodex[n]h of the triangulation Th[n]. This hypothesis implies the existence of “robust” interpolation estimates [4, 8, 13].
For any subset ω ⊆Ω, let Nh[n](ω) be the set of the vertices xof the triangulation Th[n] such thatx∈ω; let ωx[n]
h be the set of the triangles havingx[n]h as a vertex. Moreover, let ˆTx[n]
h be a triangle fromωx[n]
h where the coefficientκT achieves its maximum inωx[n]
h . We recall the following definition ofquasi-monotonicityforκ(x) from [13], referring to this reference for more details.
Definition 3.1(Quasi-monotonicity). The distribution of coefficientsκT,T ∈ωx[n]
h is said to bequasi-monotone with respect to the nodex[n]h ∈ Nh[n]
Ω
if for each triangleT ∈ω
x[n]h there exists a Lipschitz set ˜ω
T,x[n]h containing trianglesT∈ωx[n]
h such that
• ifx[n]h ∈Ω, thenT ∪Tˆx[n]
h ⊆ω˜T,x[n]
h andκT ≤κT,∀T∈ω˜T,x[n]
h ;
• ifx[n]h ∈∂Ω, thenT ⊆ω˜T,x[n]
h ,∂ω˜T,x[n]
h ∩∂Ω>0 andκT ≤κT,∀T∈ω˜T,x[n]
h .
Let the distribution of coefficientsκT,T ∈ Th[n] bequasi-monotonewith respect to every pointx[n]h ∈ Nh[n]
Ω . For a triangleT ∈ Th[n] and an edgeE ∈ Eh,Ω[n], beingEh,Ω[n] the set of the edges of the triangulationTh[n], let us define two sets containing some neighboring triangles
˜
ωT[n]=
x[n]h ∈Nh[n](T)
˜ ωT,x[n]
h , ω˜[n]E =
x[n]h ∈Nh[n](TE)
˜ ωT,x[n]
h ,
whereTE is the triangle of the two triangles sharingE whereκT achieves the maximum.
Definition 3.2 (Quasi-interpolation operator). [4, 8, 13] Let the distribution of coefficients κT, T ∈ Th[n] be quasi-monotone, then we define the quasi-interpolation operatorIh : V →V[n]h as
Ihv=
dimNh[n](Ω) i=1
λi(x)px[n]
h,i, px[n]
h,i = 1 Tˆx[n]
h,i
Tˆ
x[n]
h,i
vdΩ, ∀x[n]h,i∈ Nh[n](Ω).
Letpx= 0 for nodal pointsx∈∂Ω.
We recall from [13] the following results:
Lemma 3.3. Let T∈ Th[n] and E∈ Eh[n] be arbitrary. Let the quasi-monotonicity condition be satisfied with respect to any node x[n]h of T andTE. Then we have the following interpolation error estimates:
v−Ihv0,T ≤Cl˜R h[n]T
√κT
T∈˜ω[n]T
√
κT∇v0,T, ∀v∈H1( ˜ωT[n]), (8)
|v−Ihv|1,T ≤Cl˜R,1 1
√κT
T∈˜ω[n]T
√
κT∇v0,T, ∀v∈H1( ˜ωT[n]), (9)
v−Ihv0,E ≤Cl˜ E
h[n]E κˆω[n]
E
T∈˜ωE[n]
√
κT∇v0,T, ∀v∈H1( ˜ωE[n]), (10)
the constants Cl˜R,Cl˜R,1 andCl˜E depending only on the smallest angle in the triangulation.
For each triangleT∗∈ Th[n−1,n]such thatT∗⊆T ∈ Th[n]we define ˜ω[n]T∗=
T∈Th[n−1,n] :T⊆T∈ω˜[n]T ⊆Th[n]
, i.e. the set of the triangles of Th[n−1,n] contained in, or equal to, a triangle T ∈ Th[n] belonging to ˜ωT[n]. Moreover if E∗ ∈ Eh[n] or if E∗ ⊂ E ∈ Eh[n], then we define ˜ω[n]E∗ =
T∈Th[n−1,n]:T⊆T∈ω˜[n]E ⊆Th[n]
, else if E∗ ∈ Eh[n−1,n]/Eh[n] and E∗ ⊂ E ∈ Eh[n] let T ∈ Th[n] be the triangle such that E∗ is inside T, then
˜ ω[n]E∗ =
T∈Th[n−1,n]:T⊆T∈ω˜[n]T ⊆Th[n]
, i.e. the sets of triangles of Th[n−1,n] contained in, or equal to, a triangle of ˜ωT[n].
Lemma 3.4. Let T∗∈Th[n−1,n] andE∗∈Eh[n−1,n] be arbitrary. Then we have the following interpolation error estimates:
v−Ihv0,T∗≤ClRh[n−1,n]T∗
√κT∗
T∈˜ω[n]T∗
√
κT∇v0,T∗=ClRh[n−1,n]T∗
√κT∗
√ κ∇v
0,˜ωT[n]∗,∀v∈H1( ˜ωT[n]∗), (11)
v−Ihv0,E∗≤ClE
h[n−1,n]E∗
ˆκω[n]
E∗
T∈˜ω[n]E∗
√
κT∇v0,T∗=ClE
h[n−1,n]E∗
ˆκω[n]
E∗
√ κ∇v
0,˜ω[n]E∗,∀v∈H1( ˜ωE[n]∗), (12)
the constantsClR, andClE depending only on the smallest angle in the triangulationTh[n]and the constantCtr. Proof. Inequality (11) follows from (8) noting thatv−Ihv0,T∗ ≤ v−Ihv0,T whereT∗∈ Th[n−1,n] ⊆T ∈ Th[n], that κT =κT∗ and applying condition (7). If E∗ =E ∈ Eh[n] or E∗ ⊂E ∈ Eh[n] inequality (12) comes
from (10), (7) and considering that ˆκω[n]
E∗ = ˆκω[n]
E . While ifE∗ ∈ Eh[n−1,n]/Eh[n] and E∗ ⊂E ∈ Eh[n] then E∗ is insideT ∈ Th[n]and we apply standard trace inequality with (8) and (9) to one of the two trianglesT∗∈ Th[n−1,n]
sharingE∗, with ˆκω[n]
E∗ =κT and condition (7) to get (12).
Let us consider the spaces H10(Ω) and H−1(Ω) respectively equipped with the norms:
v2κ,1=√ κ∇v2
0=
Ω
κ(x)∇v· ∇vdΩ, Fκ,−1= sup
v∈H10(Ω)
F , v vκ,1·
Let us define the error of our approximationuh,∆t in the intervalI[n] as eh,∆t=uh,∆t−uand define ∂ u∂th,∆t =
u[n]h,∆t−u[n−1]h,∆t t[n]−t[n−1] .
Definition 3.5. Let us define the residuals in the trianglesT∗∈ Th[n−1,n]and inter-element jumps on the edges E∗∈ Eh,Ω[n−1,n] of our approximationuh,∆t
R[n]T∗ = ∂ uh,∆t
∂t −θ κT∗u[n]h,∆t−(1−θ)κT∗u[n−1]h,∆t −θΠTf[n]−(1−θ) ΠTf[n−1]
T∗
,
R[n]Ω =
T∗∈Th[n−1,n]
R[n]T∗, JE[n]∗ =
θκT
∂ u[n]h,∆t
∂nE∗ + (1−θ)κT
∂ u[n−1]h,∆t
∂nE∗
E∗
.
Definition 3.6. Let us define the following local-in-space local-in-time estimators
ηR,T[n] ∗
2
= ∆t[n]
⎛
⎜⎝ h[n−1,n]T∗
2 1
√κT∗ R[n]T∗
2
0,T∗
+ 1 2
E∗∈E(T∗)∩ Eh,Ω[n−1,n]
h[n−1,n]E∗
1
κˆω[n]
E∗
JE[n]∗
2
0,E∗
⎞
⎟⎠,
η∇,T[n] ∗
2
= ∆t[n]√
κT∗∇
u[n]h,∆t−u[n−1]h,∆t 2
0,T∗.
Then, we define the following global-in-space and local-in-time estimators
ηR[n]
2
=
T∗∈Th[n−1,n]
η[n]R,T∗
2 ,
η∇[n]
2
=
T∗∈Th[n−1,n]
η∇,T[n] ∗
2 ,
η[n]f,θ,∆t[n]
2
= t[n]
t[n−1]
ΠTf−θΠTf[n]−(1−θ) ΠTf[n−1]2
κ,−1dt,
ηf,Π[n]
T
2
= t[n]
t[n−1]
f −ΠTf2κ,−1dt,
ηf[n]
2
=
η[n]f,θ,∆t[n]
2 +
ηf,Π[n]
T
2 .
In the sequel we will derive upper and lower bounds for the error involving the following norm:
|||eh,∆t|||κ,I[n]= t[n]
t[n−1]
∂ eh,∆t
∂t 2
κ,−1
dt+ t[n]
t[n−1]
eh,∆t2κ,1dt 12
.
Remark 3.7. Following considerations of [14, 17], we can say thatη[n]R is a space error estimator related to the triangulationTh[n], whereasη[n]∇ gives information on the error due to time discretization.
Remark 3.8. The quantity ηf[n] is an estimator of the data approximation error and can be split into two terms: ηf,Π[n]T, that gives information essentially on the space data approximation error and ηf,θ,∆t[n] [n], that is a time data approximation error.
3.2. Upper bound
Theorem 3.9. Under the assumptions on the continuous problem (4)and on the discrete formulation (5), for eachn= 1, ..., N, there exists a constant C˜[n−1][n] independent of any meshsize, timestep, problem-parameter and depending only on the smallest angle of the triangulation Th[n], on the constant Ctr and on the parameter θ, such that
u[n]h,∆t−u[n]2
0+ t[n]
t[n−1]
√
κ∇eh,∆t2
0dt≤u[n−1]h,∆t −u[n−1]2
0+ ˜C[n−1][n]
η[n]R
2 +
η[n]∇
2 +
η[n]f
2
. (13) Proof. Let us define
Eh,∆[n]t= t[n]
t[n−1]
∂ eh,∆t
∂t , eh,∆t
+ (κ∇eh,∆t,∇eh,∆t)
dt (14)
and
Eh,∆[n]t=1 2
u[n]h,∆t−u[n]2
0−1 2
u[n−1]h,∆t −u[n−1]2
0+ t[n]
t[n−1]
√
κ∇eh,∆t2
0dt . (15)
From the continuous variational formulation of problem (4) it immediately follows that
Eh,∆[n]t= t[n]
t[n−1]
∂ uh,∆t
∂t , eh,∆t
!
+ (κ∇uh,∆t,∇eh,∆t)
dt− t[n]
t[n−1]
∂u
∂t, eh,∆t
+ (κ∇u,∇eh,∆t)
dt .
Recalling (5) withIheh,∆t∈Vh[n] as test function, we get
Eh,∆[n]t= t[n]
t[n−1]
∂ uh,∆t
∂t , eh,∆t−Iheh,∆t
! +
θκ∇u[n]h,∆t+(1−θ)κ∇u[n−1]h,∆t ,∇(eh,∆t−Iheh,∆t)
−
θΠTf[n]+(1−θ) ΠTf[n−1], eh,∆t−Iheh,∆t
"
dt +
t[n]
t[n−1]
θ
κ∇
uh,∆t−u[n]h,∆t
,∇eh,∆t
dt+ t[n]
t[n−1]
(1−θ)
κ∇
uh,∆t−u[n−1]h,∆t
,∇eh,∆t
dt
− t[n]
t[n−1]
(f−ΠTf , eh,∆t)dt− t[n]
t[n−1]
ΠTf−θΠTf[n]−(1−θ) ΠTf[n−1], eh,∆t
dt . Now we definet[θ,n]=θ t[n]+(1−θ)t[n−1] and we note that
uh,∆t−u[n]h,∆t = t−t[n]
t[n]−t[n−1]
u[n]h,∆t−u[n−1]h,∆t
, (16)
uh,∆t−u[n−1]h,∆t = t−t[n−1]
t[n]−t[n−1]
u[n]h,∆t−u[n−1]h,∆t
, (17)
θ
uh,∆t−u[n]h,∆t
+ (1−θ)
uh,∆t−u[n−1]h,∆t
= t−t[θ,n]
t[n]−t[n−1]
u[n]h,∆t−u[n−1]h,∆t
. (18)
