Recovery Procedures in Error Estimation and Adaptivity: Adaptivity in Linear Problems

O.C. Zienkiewicza; B. Boroomanda; J.Z. Zhub    a University of Wales, Swansea, UK
b U.E.S. Inc. Annapolis, USA

1 INTRODUCTION

Two types of procedures are currently available for deriving error estimators. They are either Residual based or Recovery based.

The residual based error estimators were first introduced by Babuška and Rheinboldt in 1978 [1] and have been since used very effectively and further developed by many others. Here substantial progress was made as recently as 1993 with the introduction of so called residual equilibration by Ainsworth and Oden [2].

The recovery based error estimators are, on the other hand, more recent having been first introduced by Zienkiewicz and Zhu in 1987 [3]. Again these were extensively improved by the introduction of new recovery processes. Here in particular the, so called, SPR (or Superconvergent Patch Recovery) method introduced in 1992 by the same authors [4]-[5] has produced a very significant improvement of performance of the Recovery based methods. Many others attempted further improvement [7]-[9] but the simple procedure originally introduced remains still most effective.

In this paper we shall concentrate entirely on the Recovery based method of error estimation. The reasons of this are straightforward:

(i) the concept is simple to grasp as the approximation of the error is identified as the difference between the recovered solution u* and the numerical solution uh; thus the estimate

e*||=||u*−uh||

in any norm is achieved simply by assuming that the exact solution u can be replaced by the recovered one.

(ii) as some recovery process is invariably attached to numerical codes to present more accurate and plausible solutions, little additional computation is involved;

(iii) if the recovery process itself is superconvergent, it can be shown [5] that the estimator will always be asymptotically exact (we shall repeat the proof of this important theorem in the paper);

(iv) numerical comparisons on bench mark problems and more recently by a ‘patch test’ procedure introduced by Babuška et al [11]-[14] have shown that the recovery procedures are extremely accurate and robust. In all cases they appear to give a superior accuracy of estimation than that achievable by Residual based methods.

It is of interest to remark that in many cases it is possible to devise a Residual method which has an identical performance to a particular recovery process. This indeed are first noted by Rank and Zienkiewicz in 1987 [15] but later Ainsworth and Oden [16] observed that this occurs quite frequently. In a recent separate paper Zhu [17] shows that:

(v) for every Residual based estimator there exists a corresponding Recovery based process. However the reverse in not true. Indeed the Recovery based methods with optimal performance appear not to have an equivalent Residual process and hence, of course, the possibilities offered by Recovery methods are greater.

In this paper we shall describe in detail the SPR based recovery as well as a new alternative REP process which appears to be comparable in performance.

With error estimation achieved the question of adaptive refinement needs to be addressed. Here we discuss some procedures of arriving at optimal mesh size distribution necessary to achieve prescribed error.

2 SOLUTION RECOVERY AND ERROR ESTIMATION

In what follows we shall be in general concerned with the numerical solution of problems in which a differential equation of the form given as:

TDSu+b=0

has to be solved in a domain Ω with suitable boundary condition on:

Ω=Γ

In above S is a differential operator usually defining stresses or fluxes as

=DSu

where D is a matrix of physical parameters.

We shall not discuss here the detail of the finite element approximation which can be found in texts [18]. In there the unknown function u is approximated as:

≈uh=Nu¯

which results in approximate stresses being:

h=DSuh=DBu¯

In above

=Nxii−1−3andB=SN

are the spatially defined shape-functions.

The solution error is defined as the difference between the exact solution and the numerical one. Thus for instance the displacement error is:

u=u−uh

and the stress error is

σ=σ−σh

at all points of the domain. It is, however, usual to define the error in terms of a suitable norm which can be written as a scalar value

=||eu||=||u−uh||

for any specific domain Ω. The norm itself specifies the nature of the quantity defined. The well known energy norm is given, for instance, as

e||E=∫Ωσ−σhTD−1σ−σhdΩ12

With the Recovery process we devise a procedure which gives, by suitable postprocessing of uh and σh, the values of u* and orσ* which are (hopefully) more accurate and we estimate the norm of the error as:

e||≈||e*||=||u*−uh||

In the case of energy norm we have:

e*||E=∫Ωσ*−σhTD−1σ*−σhdΩ12

The effectivity index of any error estimator is defined as:

=||e||este

or in the case of recovery based estimators:

*=||e*||e

A theorem proposed by Zienkiewicz and Zhu [5] shows that for all estimators based on recovery we can establish the following bound for the effectivity:

−e˜e<θ*<1+e˜e

In above e is the actual error (viz Equation (10) and (11)) and ˜ is the error of the recovered solution i.e.

e˜||=||u−u*||

The proof of the above theorem is straight forward, if we rewrite Equation (12) as:

e*||=||u*−uh||=||u−uh−u−u*||=||e−e˜||

Using now the triangle inequality we have:

e||−||e˜||≤||e*||≤||e||+||e˜||

from which the inequality (16) follows after division by ||e||. Two important conclusions follow:

(1) that any recovery process which result in reduced error will give a reasonable error estimator and, more importantly,

(2) if the recovered solution converges at a higher rate than the finite element solution we shall always have asymptotically exact...