Superconvergence for rectangular mixed finite elements

SummaryIn this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL2 between the approximate solution and a projection of the exact one is of higher order than the error itself.This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.


Introduction
We consider the approximation by a mixed finite element method for the elliptic problem where 12oR 2 is an open domain which can be decomposed into the union of rectangles, a(x) is a Lipschitz function such that 0 < ~ < a (x)< M in f2, fe L 2 (12) and g~H1/2(~O). The mixed method uses a weak formulation of the first order system (1.2) q = -a Vu div q =f to approximate q (usually a variable of physical interest) and u separately. Several finite element spaces for that formulation have been introduced [4,5,19] and convergence in different norms have been extensively analyzed I-3, 19,9,16,11]. Also the phenomenon of superconvergence has been analyzed in several papers 1-7, 8, 2, 16, 13].

R. Durhn
Here we restrict our analysis to the case of rectangular elements. In this case we prove that the distance in L 2 between the approximate solution qh and a suitable projection of q (the so called Fortin's interpolation) is of higher order than the L 2 norm of the error.
For the sake of simplicity the proofs are given for the spaces of Raviart and Thomas [,19] but the result is also valid for those of Brezzi et al. [-4] as it is shown in the last section.
This result is exploited in the lowest degree case to define a simple postprocess to obtain a higher order approximation of the vector variable q. This postprocessing could be used to construct a posteriori error estimators as was done for the standard finite element method [1,12].
Also we obtain superconvergence at Gaussian lines. Similar results have been obtained recently in [,13] using different arguments. However our approach gives superconvergence for the vector variable with optimal regularity requirements, thus, improving the results in [,13]. Moreover the domain need not be a rectangle and the estimates depend only on norms of q (and not of u) usually a better behaved variable.
An outline of the rest of the paper is as follows. In Sect. 2 we recall the mixed formulation and introduced the finite element spaces and projections.
In Sect. 3 we prove the main result concerning the error between the approximate solution and a projection of the exact one. In Sect. 4 we introduce and analyze a postprocess for the lowest order degree case. Section 5 deals with superconvergence at Gaussian lines; and finally, in Sect. 6, we comment about the extensions of the results to the spaces of [-4] as well as to the three dimensional case.
As was shown in [14,15,9,11] the error analysis can be simplified using a projection operator,
To prove it, let us first show that in this case, because of the orthogonality of the Legendre polynomials.
In the same way we can prove that for every side l of R, again because of the orthogonality. Therefore, all the degrees of freedom (2.6) and (2.7) defining HRp are zero and so (3.2) is verified.

div(qh--Flh q)=O
and so we can take r = qh-Hhq in (3.7) obtaining the theorem.

Postprocessing
In this section we apply Theorem 3.1 to obtain a higher order approximation of q by a simple local postprocessing of qh. For the sake of simplicity we consider the lowest degree case (i.e. k=0) but similar ideas can be applied for a general k.
The new approximation will be Kh(qh ) where Kh is defined as follows.
Our next goal is to prove that Kh(qh) is a higher order approximation of q than qh. llr-Kn, e(r)llo, g,,< C h21rl2,R,, for any interior node P. Now

PeN
As it is known that,  Proof. For R ~ Jh we have from the definition of K h,

ItKhrIIL~R) < C Ilrl[LOOtR)
where /~ is the union of those elements having some common vertex with R. Therefore, a local version of (4.6) is obtained using an inverse inequality and the Lemma is proved. Proof. Given a side I of an element R we have, S Ilhr'ntdl= ~ r'nzdl" 1 1 So, if re l,Pt (R)] 2 we can use the midpoint rule to obtain,
In this section we apply the results of Sect. 3 to obtain analogous estimates to those of [ 13] but when q e [H k + 2 (f2)]2 (i.e. optimal regularity).
Given an element R = [a, b] x [c, d] we use the same notation as in Lemma 3.1 for the Legendre polynomials. We denote by g~, ..., gk+l the Gaussian points in I-a, b] (i.e. the roots of lk+l) and by ~1, ..., gk+ ~ the Gaussian points in [c, d].
As in [13]  Indeed, proceeding as in Lemma 3.1 it is enough to verify it when r =(ctTk+ I(Y), fllk+ 1 (x)). As it was shown in that temma in this case we have and therefore, Therefore, in view of (5.5) and (5.6) we can apply the Bramble-Hilbert lemma to get, IIIq-/-/hq III < Chk+2 Iql~+2 which together with (5.3) and (5.4) proves the theorem.

Extensions to other spaces
In [4] Brezzi et al. introduced other spaces of mixed finite elements based on rectangular decompositions. The advantage of these spaces is that they provide the same order of accuracy than those of Raviart and Thomas but with fewer degrees of freedom.
Given (Note that for k = 0 this space coincides with the Raviart-Thomas space).

R
In view of these definitions all the arguments of Sect. 3 can be repeated for this case. Also the results on superconvergence at Gaussian points can be extended straightforward to this case.
For the three dimensional case, natural extensions of the Raviart-Thomas spaces were introduced by Nedelec [18] and new spaces (having the same advantage than their analogous in the two dimensional case) were defined in [4].
As it is easily seen, all our results can be extended to these cases.