On the asymptotic exactness of error estimators for linear triangular finite elements

SummaryThis paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.


I Introduction
In recent years, considerable interest has been shown in a posteriori error estimates and adaptive refinement for finite element approximations of second order elliptic problems.
In the one variable case, a rather complete theory has been developed by Babu~ka and Rheinboldt [5,7,8].In particular, they have proven that, under suitable regularity assumptions on the solution, several estimators are asymptotically exact in the energy norm.The estimators they have considered are essentially of two kinds.The first one is based on the solution of local problems while the second one, on the computation of residuals and jumps of the approximate solution.
These ideas have been generalized to problems in two variables [-6, 3, 9-1, but the analysis is much more complicated in this case.
////// ////// ////// Fig. 1 Estimators based on the jumps of the normal derivative of the approximate solution can be constructed and analyzed by means of superconvergence results.In particular, for rectangular elements, the results of Zlfimal [25] can be used to device several error estimators [1,12], and moreover to show that they are asymptotically exact [-12].These estimators can be viewed as generalizations of a one-dimensional estimator of Babu~ka and Reinboldt [8].
Here we consider the case of linear triangular elements.Superconvergent recovery of the gradient for particular meshes has been considered by several authors.A fundamental result was obtained by Oganesjan and Ruhovec [19] for uniform meshes (like those in Fig. 1) that states that the difference between the gradient of the approximate solution Uh and the gradient of the Lagrange interpolation u ~ of the exact solution u is of higher order than the error itself whenever the solution is regular enough, that is, (1.1) IIV(uh--ul) ll 0,~= O( hl +E) for some e > 0. We say that there is superconvergence whenever (1.1) holds.
In order to construct an asymptotically exact error estimator, we define a recovery operator for the gradient, that is, an operator G such that Guh is a higher order approximation of Vu than VUh itself; namely,

IlVu-Guhllo.o=O(h x §
for some e > 0. Given G, we define the error estimator e by e,=Guh-Vu n.
In the case of a uniform mesh like that in Fig. 1, an operator G based on the interpolation of the average of the two gradients in the triangles sharing a common edge was introduced in 1-24].
In this work, we introduce a recovery operator G based on quadratic isoparametric interpolation.We show that for subregions f20~ 01 c f2 in which the mesh satisfies a regularity assumption (we call it quasi-parallelism and, intuitively, it means that the meshes are higher order perturbations of uniform meshes).The superconvergence result (1.1) is also valid for quasi-parallel meshes [23], therefore, the error estimator based on G is asymptotically exact in the sense that the relative error (1.4) [IE-Vello.~o,0 IIVello,~o h~O whenever the Hl-seminorm of the error e:=u-u h is properly O(h).(Note that (1.4) implies asymptotical exactness as defined by Babu~ka and Rheinboldt [8].) Afterwards, we calculate the value of Gu I at each edge midpoint and we show that it is a weighted average of the restrictions of Vu I to the triangles containing that point.So, by interpolating linearly those values in each triangle, we obtain another operator G which is a generalization of the operator introduced in [24] for uniform meshes.
We also prove that and so, defining g:=dUh--Vuh, we obtain a result analogous to (1.4) for this estimator.
The estimator g can be easily computed in terms of the jumps of the normal derivative of the approximate solution.Estimators of this kind are actually in use [14,18,20,22].
On the other hand, the techniques by Babu~ka and Miller [4] can be applied for any general triangular mesh, regular in the usual sense, and for any problem whose solution ueH 1 (f2), to show that the estimator g is equivalent to the error in the energy norm; namely, there exist constants C1 and C2 such that C1 II~ll o,~ IlVel{o.~C2 II ~11o.~.
Therefore, it is natural to ask if the asymptotic exactness is satisfied in regions where the meshes are not quasi-parallel.We show that this is not true.Specifically, we consider a subregion where the meshes are of the so called criss-cross type.For the case of Laplace equation, numerical evidence of the lack of superconvergence for this kind of meshes was presented by Levine [16].We give here a proof of this fact and, by showing that (1.5) holds in these meshes for a family of functions, we provide examples where g is not asymptotically exact.
Finally we analyze some particular problems and show that the asymptotical exactness of g fails even in cases where there is superconvergence.Also we use these examples to show that g is not asymptotically exact even in the weaker sense of Babu]ka and Rheinboldt [8].As a conclusion, we can say that the assumption on the meshes is not only a technical matter; that is, the asymptotical exactness of g does not hold in general.
The remainder of the paper is organized as follows.In Sect. 2 we introduce the model problem and notations; Sect. 3 deals with recovery operators and error estimators, and finally in Sect. 4 we analyze the case of criss-cross meshes.Let {~} be a regular family of triangulations of ~, where, as usual, h stands for the mesh size.Let u h e Vh"= {V e H 1 (f2): v Ire ~1, V Te ~} be the piecewise linear finite element approximate solution of problem (2.1).(~m denotes the set of polynomials of degree not greater than m).Let e:=u-uh denote the error of this approximation.
From now on C will denote a constant independent of h and u, but not necessarily the same at each occurrence.

Recovery operators and error estimators
First, we define a recovery operator based on quadratic isoparametric interpolation.In order to do this, let us introduce some notation.Given an interior element Te~ we denote by T~, i= 1, 2, 3 its neighbor triangles.Let K and Ki be reference triangles as in Fig. 2. We set  We want to define a recovery operator G such that Gu ~ be a better approximation to Vu than Vu ~.For VeVh we define Gv locally in such a way that GriT takes into account only the information of the six nodal values v(Ai), i=1 .... , 6.In order to make clear the idea behind the definition of G, let us first remark that in the simpler case that T=K and T~=Ki, i= 1, 2, 3, we would define for any v ~ Vh and x ~ K,
To extend this definition of G to the general case, we need to assume that F: K*---,IR 2 is a one to one correspondence with a smooth inverse defined on T.'=F(K*).In such a case, for ve Vh we first define for x~ T*, where z3=vo* e, and then for xeT,

Gv(x):=Gr v(x).
For a boundary triangle T we choose an interior element S neighbor of Tand repeat the construction using T*..=S* to define Grv'.=Gsv.
In order to prove the asymptotic exactness of the estimator defined by means of this recovery operator, we need to assume a further regularity assumption on the meshes.
Given a subregion O1 c f2 we say that the meshes of the family {~h} are quasi-parallel on f2~ if for any triangle TeJh such that Tcf2~, and for any neighbor T'e3--h sharing an edge with T (see Fig. 3), the opposite angles of the quadrilateral T~ T' differ in O(h).That is, according to the notation of Fig. 3, I~-al =O(h),

Ifl-Tl=O(h).
An analogous assumption is that if D' is the point such that ABCD' is a parallelo- It is easy to show that quasi-parallelism is equivalent to the usual regularity for the isoparametric quadratic triangle T.Under this hypothesis,/~ has a smooth inverse (see [10]).Therefore, G is well defined for quasi-parallel meshes but this is not a necessary assumption.In fact, exception made of some degenerate meshes, ff will have always a smooth inverse and hence G will be well defined.
On the other hand there are several different assumptions implying quasiparallelism that have been used in previous works about superconvergence in linear triangular elements; for instance those in [16] and [23].In particular, whenever the nodes of the meshes in ~h are images of the nodes of uniform meshes under a fixed diffeomorphism, they are quasi-parallel.
From now on, we assume that we have a family of quasi-parallel meshes on a subregion f21 C f2 and we shall prove some properties concerning the recovery operator G. Proof.Since 12 (~i)= 12 f3 we have, NOW, Let 0o~21 be subdomains of f2 and assume as before that the mesh is quasiparallel in f21.It has been proven [23] that, for h small enough, For a convex polygon, if the solution ueH2(O), then it is easily seen that (3.9) holds.For a nonconvex polygon, (3.9) also holds for instance for the Laplace operator withfeL2(f2) and homogeneous Dirichlet boundary conditions.In fact, Fig. 4 in such a case, the solution of (2.1) belongs to H~+S(O) for all s<Tt/co where o) is the largest angle of the polygon (see Grisvard [13]) and Ilello,m <ChZ~lula+s,o.
From now on we assume that (3.9) holds and also that the error is properly O (h) in f2o ; that is, (3.10)

lel l,ao > C h
for some constant C depending on u but not on h.This last requirement is not very restrictive, being satisfied in all but trivial cases (see [4]).
Therefore, if the solution is in H3(Q1), then the estimator e is asymptotically exact in f2 o, that is, II e-Ve I10.ao _ O(h~).

IlVeHo,~o
In order to define a simpler error estimator, let us compute the values of Gv, for ve Vh, at the midpoint of any interior edge.
Applying the chain rule to (3.1) we have Now let us observe that for any quadratic function g defined on K*, Vg(Qi)= V(gIIK) + V(g'lK). 2 hence and also
In the simpler case in which F is affine (and hence F=i~), (3.11) holds.In fact,

GT v(Qi)= 2
In view of this, we may assume that T= K and restrict ourselves to consider the case described in Fig. 5 for any aeP,.and fl>0.
In this case, a straightforward computation yields and so, by using (3.12) we conclude the lemma.[] In particular, (3.11) shows that Gv is continuous at those midpoints.This fact allows us to define a second recovery operator: by interpolating the values of G v at the midpoints of the sides by a piecewise linear function, that is, and where Jg is the set of midpoints of all the edges of the mesh.Now, using Lemmas 3.4 and 3.5 we may conclude that if the solution is regular enough then the estimator ~" is asymptotically exact.That is, if uffH3(~21) and if (3.9) and (3.10) hold, then

II~'-Vello,~o = O(h~).
NVello,oo Remark 3.3 By using the techniques developed by Babuska and Miller [4] it is possible to prove that the estimator ~" is equivalent to V e in the L2-norm for very general meshes and without any extra assumption of regularity of the solution.Namely, if the family of meshes Jh is regular in the sense of [10], ueH 1 (f2) and (3.10) holds, then there exist two constants C1 and C2 such that

Non asymptotic exactness of
Estimators like ~ based on the jumps of ~ are widely used in practical computations (v.g.[14,18,20,21,22]).We have just proved that ~ is asymptotically exact in those regions where the meshes are quasi-parallel; therefore, it arises naturally the question of whether or not this extra regularity of the triangulations is essential.
In this section, we analyze the behavior of the estimator ~ in regions where the meshes are very regular but not quasi-parallel.Specifically, we consider Fig. 6 a region where the meshes are of the criss-cross type and uniform (see Fig. 6) and the differential operator is the Laplacian.
Numerical evidence of the lack of superconvergence for this kind of meshes was presented by ].First, we give here a proof of this fact.Let us denote by R the square in Fig. 6.In what follows, we use the notation in that figure.
for any w~2.Since q~=0 on 8R we have On the other hand, where ~h is the unit normal to the side [P, PJ as in Fig. 6 where we used that ve~2.Therefore, subtracting (4.3) from (4.2) we have and so Lv=O for any yeN2.On the other hand, by using that HVtplIO,R=2, the usual interpolation theory and the Cauchy-Schwarz inequality we have R Therefore, a standard application of Bramble-Hilbert Lemma (see for instance [11]) yields for any vffH3(R) thus concluding the proof.[] Theorem 4.1.Let f2 o c f2 be a region where the mesh is like in Fig. 6.Let u e H 3 (Oo) be such that I ~ Aul>ct fo r some constant c~>0.Then, f2o hct Ch z Proof.Let RcOo and q)eVh as in the previous lemma.Then S (Vu.-W').v~=S (vu~-Vu').v~o=~ (w-vu').v~oThis theorem shows that on any subregion where the mesh is of the criss-cross type and for problems with rather general solutions, there is no superconvergence in the sense of (1.1).
In order to apply Theorem 4.1 to the analysis of the estimator ~ we also need the following property.
Some simple calculations show that for any ve Vh, where Qi is the midpoint of the side I-P, PJ, and

So the lemma holds. []
We want to exhibit some particular problems for which g is not asymptotically exact.Note that this is not an immediate consequence of the previous lemma since for nonquasiparallel meshes ~(u I) is not necessarily a superconvergent approximation of Vu.In the next lemma, we estimate the difference Vu-d~(u~t).Let us define an operator L: H3(T*)~EI_?(T)] 2 by Lv:=Vv-GvI-Ev.First, we prove that Lv=O for any ve~'2.Indeed, in this case Lve(~l x~l) and so it is sufficient to prove that Lv vanishes at P, Q and R.

=h wv(ll).
Moreover, Cv = h Wv and so, computing the values of Ev, we see that (V v -G v t) and Ev coincide at P, Q and R. Therefore, Lv=O for ve~2.So, we have shown that the estimator g is not asymptotically exact in general.However, there are still two natural questions.
i) Is ~" asymptotically exact whenever there is superconvergence in the sense of (1.1)?
ii) It is asymptotically exact in the weaker sense of [8]? I.e.: does the so called effectivity index 0r~o,= II~llo,~o ,1'~ I]Vello,~o h~o " We are going to show that the answer to these two questions is negative.In order to do that, let us consider a problem such that the solution ue~2 and meshes which are criss-cross in the whole O.In this simple case we may calculate explicitly u h.
For each square R as in Fig. 6 the equation corresponding to the node P gives:  Considering a patch like that in Fig. 8 and eliminating the unknowns corresponding to the middle point of each square we obtain: Therefore, the difference scheme associated with the linear elements in a criss-cross mesh is an average of two five-points finite difference schemes, which, as it is well known, are exact for u~2.Hence, for each square R (with the notation of Fig. 6) we have uh(P~)=u(P~), i=1,2,3,4, and, by using (4.7), we obtain uh(P)=~7(8u(P)+i~ 1 u(Pi))=u(P)+~ h2 Au.
Using these values of uh we obtain (with the notation of Fig. 7) On the other hand, using (4.5) for w= Uh, it is easily seen that: (4.9) Therefore, for any subregion I2 o as in Fig. 6, 0~o=~22~ 1, independently of the mesh size h.Note that in this case there is superconvergence (moreover Uh--= Ul); however the estimator g is not asymptotically exact.Secondly, for u (x, y)= xZ+ yZ, we have h2]/~ h 2 II~[Io,T = 6l//~ , and IlVe[10,T-]/~, so, 0Oo = V~ Therefore, in both examples the estimator ~ is not asymptotically 6 " exact even in the weaker sense of [8].

1)~ p~ 1 )
11 0,~,T <[TI1/21I2~oP-112,~,~, sup[x-Po~e-l(x)[.x~T But, under our hypothesis it is easily seen that, suplx-P o F-l (x)l < C h z xET and using an inverse inequality for 12 f)o/~-1 we obtain from (3.8) that the second term on the right hand side of (3.7) is bounded by Ch2112~~ 2 [I vl12,~-,where the last inequality follows from the isoparametric interpolation theory[10].Finally, to bound the first term on the right hand side of (3.7) we use again the known results for isoparametric interpolation and we obtain(3.6).[] Let us now define the error estimator e:=GUh--Vuh.

Lemma 3 . 3 .
Denoting by T-and T § two adjacent triangles and Q the midpoint of the common side, for any ve Vh we have,

Lemma 4 . 1 .
Let veH3(R) and cpe Vh be the basis function corresponding to the node P, then

h2 h 2 I
~uh(R)= u(F)--u(B) Now, let us consider two examples.First, for u(x, y)= x 2 _y2, we have h [~l[o,r-l/~, and IlVello,7 where we used a trace theorem to bound the last term.Now, a standard application of the Bramble-Hilbert Lemma yields, Collecting all the lemmas we have the following theorem which says that, in general, g is not asymptotically exact in the sense that the relative error does not tend to zero.
Note that in this case G(ul)=Vu.