Review: A posteriori error estimation techniques in practical finite element analysis

TL;DR: The basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem are reviewed and it is concluded that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care.

About: This article is published in Computers & Structures.The article was published on 2005-01-01 and is currently open access. It has received 207 citations till now. The article focuses on the topics: Human error assessment and reduction technique & Finite element method.

Summary

1. Introduction

• The modeling of physical phenomena arising in engineering and the sciences leads to partial differential equations in space and time, expressing the mathematical model of the problem to be solved.
• Since the late 1970s several strategies have been developed to estimate the discretization errors of finite element solutions.
• This new development is commonly referred to as goaloriented error estimation since the aim is to provide error estimates and error bounds for particular quantities of interest.
• Hence, error bounds can be guaranteed but still be inaccurate, whereas an error estimate should be accurate although, in general, it over- or underestimates the true error.
• The crucial question is whether these procedures are also effective in practical solutions such as in the linear analysis of geometrically complex 2D, 3D and shell problems, and in the analysis of problems including nonlinear effects, time-dependent loads or multi-physics phenomena.

2. Requirements for an error estimator

• The main purpose of any a posteriori error estimator is to provide an estimate and ideally bounds for the solution error in a specified norm or in a functional of interest if the problem data and the finite element solution are available.
• Ideally, the error estimator should yield guaranteed and sharp upper and lower bounds of the actual error.
• Of course, a key requirement for this error estimator to be useful in engineering practice is that the computational cost of the error estimate (and ideally the error bounds) must be much smaller than the added computational cost to simply use a very fine mesh.
• Today, even when considering only linear analysis, the authors are far from the ideal-error-estimator-solution, since either error bounds for quantities of interest are guaranteed but expensive to compute or not computable at all, or they are computable but not guaranteed.
• Note that global error estimates for the energy norm consider only the error in the global energy norm and do not provide any local information.

3. Model problem

• The boundary value problem consists of finding the solution u that satisfies.
• While the authors consider here a model problem with a scalar for solution, the concepts given in the paper are, of course, also applicable to the general linear elasticity problem by using the appropriate vectors of solution variables and the corresponding solution spaces.
• Indeed, in the example solutions in Section 9, the authors only consider more general elasticity problems.

3.1. Finite element approximation

• The partition, or simply mesh, formed by the union of all elements, is assumed to coincide exactly with the domain X and any two elements are either disjoint or share a common edge.
• Here, Rhð Þ is called the residual functional or the weak residual.
• Assuming that the bilinear form is positive definite it follows that kRhkV 0 ¼ sup v2V ðXÞ j RhðvÞ j kvkE ¼ kehkE ð11Þ where kRhkV 0 denotes the norm of the residual in the dual space V 0(X).

3.2. A priori error estimates

• A priori error estimates provide useful information on the asymptotic behavior of the approximation.
• Basically, Céa s Lemma asserts that the error of the finite element solution measured in the V-norm is of the same order as the interpolation error.
• Choosing V H1 and employing estimates it turns out that the error measured in the H1-norm is of the order O(hp) ku uhkH1ðXÞ 6 chpkukHpþ1ðXÞ ð14Þ where c is a stability and constant which does not depend on the actual Ansatz space 1 and h denotes the maximum of all element sizes.
• Furthermore, the authors have for the error in the L2-norm ku uhkL2ðXÞ 6 chpþ1kukHpþ1ðXÞ ð15Þ which means that the convergence rate for the solution itself is O(hp+1) [1].

4. Global error estimates for the energy norm

• The authors present various error estimators for the global error in the energy norm of the above mentioned elliptic model problem when a specific (not very fine) mesh has been used.
• Of course, it is always possible to solve problems (8) or (9) very accurately using a very fine mesh and then the exact error can be calculated.
• In order to evaluate the accuracy of any error esti- mate, the authors use the effectivity index defined by k ¼ EhkehkE ð16Þ with Eh denoting an estimate for the error in the energy norm.
• In contrast, implicit error estimators require the solution of auxiliary local boundary value problems.
• A third class of error estimators are the recovery-based error estimators.

4.1. Explicit error estimators

• Explicit error estimators involve a direct computation of the interior element residuals and the jumps at the element boundaries to find an estimate for the error in the energy norm, see the fundamental work of Babuška and Rheinboldt [9,11], Babuška and Miller [10] and Kelly et al. [12].
• Applying the Cauchy–Schwarz inequality elementwise yields aðeh; vÞ 6 X K2Th kRkL2ðKÞkv IhvkL2ðKÞ þ X c2oTh kJkL2ðcÞkv IhvkL2ðcÞ ð23Þ.
• The authors also note that on interelement boundaries oK 6 C the jump J is multiplied by the factor 1/2 to distribute the error equally onto the two elements sharing the common edge.
• Research has been conducted to evaluate these constants for specific problems—considering more general cases than their model problem—but the values obtained are related to worst-case scenarios and the error bound is generally not sharp.
• One strategy is to enrich the finite element space if the relative error erel exceeds a specified tolerance, say ctol = 0.01.

4.2. Implicit error estimators

• Implicit error estimators involve the solution of auxiliary boundary value problems whose solution yields an approximation to the actual error.
• The interest in implicit schemes stems from the fact that in explicit schemes the whole information for the total error is obtained only from the given solution, when it might be possible to obtain more accurate information on the error by solving additional auxiliary problems.
• The boundary value problems to be solved are local, which means that they are posed either on a small patch of elements (subdomain residual method) or even only on one single element (element residual method).
• In general, a drawback of the subdomain residual method can be that solving the local problems is rather expensive, since each element is considered several times.
• On the other hand, the element residual method needs to approximate the prescribed Neumann boundary data on each single element.

4.2.1. Element residual method

• On the Dirichlet part of the boundary the contribution to the local error is zero.
• Clearly, on the Neumann part of the global boundary, the true flux in (32) equals the prescribed data g.
• Unfortunately, the existence and uniqueness of the variational problem (35) is not guaranteed due to the possible incompatibility of the prescribed Neumann data.
• To overcome this drawback several techniques have been proposed.
• If not, the estimator could underestimate the error [6], but in [16] it is shown that the error arising in the approximation of the local problems can be estimated with explicit estimation schemes.

4.2.2. Subdomain residual method

• The basic idea of the subdomain residual method is to decompose the global residual Eq. (17) into a number of local problems on small element patches with homogeneous Dirichlet boundary conditions [9,11,17].
• In the approach of Prudhomme et al. [21], the subdomain residual method starts with the fundamental error representation (17) and utilizes the partition of unityproperty of the shape functions.
• Consider that in their model problem for a mesh of n nodes, the authors use n element patches, each node defining as its patch the elements coupling into the node.
• The second term on the right-hand side of (45) should decrease as the polynomial degree used to solve the local problems increases.
• In [21] the error estimator was tested in some numerical experiments.

4.3. Recovery-based error estimators

• Recovery-based error estimators make use of the fact that the gradient of the finite element solution is in general discontinuous across the interelement boundaries.
• Here, the underlying idea is to post-process the gradient and to find an estimate for the true error by comparing the post-processed gradient and the nonpost-processed gradient of the approximation.
• The attribute ‘‘superconvergent’’ is adopted from the literature, but it is somewhat misleading since it does not provide any information about the distance between the exact solution and the post-processed finite element approximation.
• Therefore, Hiller and Bathe [29] propose an element-based error estimator that uses higherorder-accuracy points to recover the strain field.
• Another weak point of the Zienkiewicz–Zhu algorithm is the implicit assumption that oscillations indicate errors and that smooth stresses mean accurate stresses.

4.4. Concluding remarks

• In order to assess the presented techniques, the authors have to realize that there are two different goals using error estimation procedures.
• The first goal is to estimate the actual error in a suitable norm and ideally provide useful actual error bounds.
• Actually, all the above mentioned error bounds when evaluated in general analyses are not guaranteed and must therefore be used with caution.
• The second goal is to steer an adaptive scheme to obtain meshes which are optimal with respect to the aim of the computation.
• Recovery-based error estimators are inexpensive too and also provide quite useful error estimates for this purpose.

5. Goal-oriented error estimates

• In finite element analysis it is frequently the case that the analyst is more interested in certain output data of the finite element approximation than in the global energy norm.
• In order to find an estimate for the error in the output data pertaining to a specific quantity, or to find at least an effective mesh to accurately solve for this quantity, error estimators for the energy norm are not useful.
• Hence, more recently so-called goal-oriented error estimates were developed, which estimate the error in individual quantities of interest using duality techniques [30–41].
• Let Q(u) denote such a quantity of interest, as for example the mean value of the x-component of the gradient in their model problem over a (not necessarily) small patch Xe, QðuÞ ¼j Xej 1 Z Xe rxudX ð58Þ.
• In the following the authors recall the basic framework of these procedures and then briefly discuss specific applications using this framework.

5.1. The basic framework

• The solution of this problem is referred to as the dual solution and can be interpreted as the generalized Green s function, or the influence function, related to the functional Q(v).
• The objective in the following is to find an estimate for the error QðehÞ ¼ QðuÞ QðuhÞ ð61Þ where Q(uh) denotes the finite element approximation of the quantity of interest and Q(u) is the exact value.
• Clearly, the more accurate the approximation zh for the influence function z, the more accurate is the calculation of the quantity of interest itself.
• Furthermore, the use of (66) in many mixed finite element methods is direct since (66) only uses the symmetry of the bilinear form.

5.2.1. Energy norm based estimates

• There are several strategies for goal-oriented error estimation based on energy norm estimates of the primal problem and the dual problem.
• Using the error representation (63) and employing the Cauchy– Schwarz inequality, the following upper error bound is obtained j QðehÞ j¼j aðz zh; u uhÞ j6 kz zhkEku uhkE ð69Þ.
• Hence, the error in the quantity of interest is bounded by the error in the energy norm of the primal problem weighted with the error in the energy norm of the dual problem.
• In [43] it is shown that this overestimation gets even worse by increasing the polynomial degree of the interpolation functions used in a p-version of the finite element method.

5.2.2. The dual-weighted residual method

• There are several strategies for evaluating the unknown dual solution in (83).
• One possibility is to solve the dual problem by using a higher-order method where, for instance, biquadratic functions are used instead of bilinear functions.
• This yields the approximate error representation QðehÞ X K2Th Z K Rðzð2Þh Ihz ð2Þ h ÞdX þ Z oK Jðzð2Þh Ihz ð2Þ h Þds ð86Þ where the upper index (2) denotes the solution obtained with a higher-order method and Ih is the interpolant on Vh.
• Both strategies lead to useful error indicators and can even lead to effectivity indices close to 1.0 depending on the problem considered.
• Error estimates using (86) or (87) may underestimate the actual error since the dual problem is only solved approximately.

5.2.3. Direct use of influence functions

• Instead of calculating element-based residuals, the authors can also directly employ influence functions and (66).
• This approach was used by Grätsch and Bathe in shell analyses [49].
• Assuming In [49] it is found in some numerical studies that the error representation (90) leads to accurate error estimates and to effectivity indices close to 1.0 (since (90) takes into account the cancellation effect of the error over the domain).
• The error estimate includes the error due to the approximation in the geometry of the shell structure.
• An example using the error representation (90) is given in Section 9.1.

5.2.4. The Green s function decomposition method

• Considering individual output data such as point values, the presented procedures face the problem that in H1(X) functions are not continuous (i.e. have meaningful point values) in two or three dimensions.
• The idea is to approximate the Green s function by splitting it into a regular part and a wellknown fundamental solution, and for this reason the method can be referred to as the Green s function decomposition method.
• Consider their model problem (see Section 3) and let u(x) denote the point quantity of interest at the point x 2 X. Comparing this result with (69), the authors realize that, to use (102), they can employ the usual error estimation techniques of goal-oriented error estimates since the error bounds in (69) and (102) only differ by the weighting factors.
• But, as mentioned already, the method is rather restrictive in that the fundamental solution needs to be available (see (97) and (98)).

5.2.5. Exact-bounds approach

• The ‘‘exact-bounds approach’’ proposed by Peraire and co-workers [57] can be used to obtain guaranteed upper and lower bounds for quantities of interest.
• The basic idea in this approach is to use a displacementbased finite element method to obtain a guaranteed lower bound on the exact strain energy and to employ a hybrid (stress-based) finite element method using complementary energy principles to obtain a guaranteed upper bound.
• Recasting the problem in terms of the dual problem corresponding to the functional of interest, guaranteed bounds can be obtained for the quantity of interest using the solutions of local problems.
• In addition, local problems have to be analyzed to obtain the error bounds, which could make the procedure costly in actual applications.
• Also, for some types of problems, notably incompressible linear and nonlinear problems and shell problems, mixed (or hybrid) formulations must be used to circumvent the ‘‘locking’’ phenomenon [1], and any approach to estimate errors needs to take this fact into account.

5.2.6. Reduced-basis output bounds approach

• The authors also want to mention some achievements which do not exactly fit into the basic framework of Section 5.1 but which also provide estimates for quantities of interest.
• The ‘‘reduced-basis output bounds approach’’ proposed by Patera and co-workers [59] addresses the practical case in which the quantity of interest needs to be computed for a certain number of parameters that are the values of specific variables describing some input data such as physical properties or geometry data.
• The underlying idea of this approach is to solve the problem for a sample of the parameters on a very fine finite element mesh.
• Then, having these solutions, the solution corresponding to any other configuration within the parameter set is obtained by some interpolation technique.
• In practice, the authors generally do not know how to choose the size of the parameter space and it certainly depends on the desired accuracy, the selected quantities of interest, and the particular problem analyzed.

5.3. Global or pollution error versus local error

• In finite element analysis the authors may ask whether it is sufficient to just use a graded and reasonably refined mesh only around the point of interest.
• This problem is similar to the problem studied by Babuška and Strouboulis [6], but the authors give further results.
• The exact solution of this model problem for different values of t and chosen data is shown in Fig.
• The results in Fig. 5 show that only in the first element the local error is larger than the global error and elsewhere the local error becomes negligible; thus, the global error dominates in all other elements.
• Indeed, in practice, the areas of high stress gradients are usually also the areas of interest and hence, natural mesh refinements in those areas frequently control the global error.

5.4. Concluding remarks

• Goal-oriented error estimation techniques focus on establishing accurate and computationally inexpensive error estimates for a quantity of interest.
• Considering the dual-weighted residual method, the error estimate may over- or underestimate the actual error if the approximate error representations (86) or (87) are used.
• Error bounds as in (70) and (85) lead generally to an overestimation of the true error, since the cancellation effect of the error is neglected [60].
• Moreover, the upper bound (85) is, strictly, not guaranteed if the unknown dual solution is approximated.
• Some results for 3D problems were recently published [61] but additional experiences need be obtained.

6. Nonlinear problems

• Any of the error estimators derived in Sections 4 and 5 may therefore also—in principle—be employed to estimate the error in nonlinear problems.
• In the following the authors explain the basic ideas for a simple model problem and emphasize goal-oriented error estimates.
• The authors employ this problem since there are no difficulties in proving the uniqueness and stability of the solution (see Ref. [62]), hence they can straightforwardly focus on the error estimate.
• In order to linearize the error representation (111) with respect to eh, the authors consider RhðwÞ ¼aðu;wÞ aðuh;wÞ ð113Þ ¼ðru;rwÞ ðu3;wÞ ðruh;rwÞ þ ðu3h;wÞ ð114Þ ¼ðreh;rwÞ ðu3 u3h;wÞ ð115Þ ¼ðreh;rwÞ ðe3h þ 3u2heh þ 3uhe2h;wÞ ð116Þ and provided that the finite element solution uh is sufficiently close to the exact solution u, they obtain RhðwÞ ðreh;rwÞ ð3u2heh;wÞ ð117Þ.
• For estimating the error in inelastic analysis, notably elastoplasticity, several techniques have been presented, see for example [64–68] and the references therein.

7. Time-dependent problems

• The commonly used finite element solution approach to solve hyperbolic differential equations is to approximate the problem with finite elements in space and to apply a finite difference scheme in time.
• To obtain the finite element solution of (133) the authors employ the standard Galerkin method with the trial and test space Vh V.
• As an example, consider the Newmark method, which is used widely [1].
• And this error can be estimated with the error procedures discussed above.the authors.
• The overall errors to represent the required frequencies and integrate accurately in time are not assessed by the procedures the authors discussed and represent major sources of errors in the solution [1].

8. Mixed formulations

• The crucial point for the solvability, stability and optimality of the finite element approximation is that the finite element spaces satisfy the discrete inf– sup condition [72–74] inf vh2V h sup sh2W h ðdivsh; vhÞ kshkW hkvhkV h P C > 0 ð149Þ.
• Note that the normal vector n and the jumps change sign if the orientation of the edge is reversed.
• To obtain error estimates for the solution of a linear quantity of interest Q(w), the authors use a slightly different procedure than in the standard approach.
• Note that the authors have kuk 6¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi.

9. Numerical examples

• The authors give some example solutions to illustrate the use of error measures.
• The authors consider in their numerical examples 2D linear elasticity and shell problems.
• The authors do not report upon actual practical engineering analyses but only give illustrative examples.
• Also, the authors are looking in some cases at very small errors, smaller than needed in practice, but do so because they want to study the convergence behavior of the procedures used.

9.1. Analysis of a shell

• Fig. 10 shows that high stress concentrations are present in two regions near the clamped boundary (corresponding to the membrane stresses) and at the tip of the structure where the loading boundary conditions change (corresponding to the bending moment).
• These quantities are evaluated using the local Cartesian coordinate system ð r; s; zÞ (see Fig. 9).
• The reference solution was obtained using a uniform mesh of 100 · 100 MITC9 elements (with 201,000 degrees of freedom) for the complete structure.
• As seen in Fig. 12 for every quantity of interest the estimated relative percentage error decreases quickly and the corresponding effectivity indices are close to 1.0.
• Then the grey regions indicate that the error in the quantity of interest is smaller than the tolerance if the load is applied there, while the white areas correspond to errors larger than the tolerance.

9.2. Analysis of a frame structure

• Fig. 14 shows the frame structure considered.
• For the solution the authors use two approaches: first, the refinement is based on the norms given in (28) and (29) and, second, the refinement is based on the goal-ori- ented error indicators given in (77)–(79).
• Fig. 15 compares the results obtained using these two solution approaches with the results obtained by simply using a uniform mesh refinement.
• In all these solutions, the quantity of interest is obtained by differentiation of the finite element displacement field to evaluate the stresses.
• These results show that, as expected, the error indicator for the von Mises stress obtained with ADINA corresponds quite well with the energy norm refinement, so that, indeed, this error indicator would have provided a good guide for driving a mesh refinement ‘‘by-hand’’ without any usage of an error estimator for the energy norm.

9.3. Analysis of a plate in plane stress

• Next, the authors study the example described in Fig. 17.
• First, the authors give some results obtained employing the Green s function decomposition method using uniform meshes.
• As a result of this uniform refinement shown in Table 1, the error in the stresses decreases reasonably fast, and indeed highly accurate results are achieved even on coarse meshes.
• Next, the refinement is steered using the global energy norm control based on the explicit error estimator in (28) and (29) with the constants given in (168), and in a third solution the refinement is obtained using the energy-norm-based goal-oriented strategy based on (77) using also the constants in (168).
• As seen, these error bounds produce an envelope which contains the exact solution, and which is quite narrow although the estimates for the local errors are based on explicit error estimators.

10. Conclusions

• In this paper the authors reviewed some basic a posteriori error estimation techniques which broadly can be classified into global error estimators for the energy norm and goal-oriented error estimators to provide error estimates and error bounds for linear quantities of interest.
• On the other hand, to actually bound the error almost guaranteed in a suitable norm is at present only possible for certain problems, and then very expensive in practical analysis.
• The results of the analysis are frequently most accurate when the relative error is uniform over the complete analysis domain.
• In order to obtain highly accurate local quantities of interest, the mesh should of course be reasonably fine around the region of interest (and also in general in the areas of high stress gradients as mentioned already).
• Hence, while the theory of error estimation has provided much valuable insight into the finite element solution process, many of the proposed techniques are at present only valuable to a limited extent in engineering practice.

Figures

Fig. 16. Results obtained in the analysis of the frame in Fig. 15: (a) In mesh obtained with the energy norm control and (c) final mesh obtai

Fig. 14. Frame under horizontal body load to study the error in the integrated shear stresses in the two sections A–A.

Fig. 15. Comparison of different adaptive refinements to approximate the integrated shear stresses in the two sections A–A.

Table 1 Stresses at A employing the Green s function decomposition approach using uniform meshes

Fig. 17. Plate in plane stress condition to study the error in the stresses at the point A: (a) problem data, (b) rxx-stress field, (c) ryystress field and (d) rxy-stress field.

Fig. 9. Goal-oriented error measures for a shell structure: problem description.

Fig. 3. Oscillating longitudinal displacements u(x) produce an oscillating normal force N(x) but the finite element solution uh(x) is zero.

Fig. 11. Goal-oriented error measures for a shell structure: locations of three quantities of interest. 20 · 20 mesh shown.

Fig. 10. Stresses of the shell structure under line load: (a) r s s-stress field and (b) r r r-stress field.

Fig. 19. Final meshes to approximate the stress rxx at point A: (a) goal-oriented refinement, (b) refinement using the Green s function decomposition approach.

Fig. 1. The ideal-error-estimator-solution in finite element analysis: guaranteed maximum absolute error bounds at every point of the structure. Not available yet.

Fig. 8. Global error for the plate in plane stress condition: (a) error in ux-displacement, (b) error in uy-displacement.

Fig. 7. Plate in plane stress condition to study the global error.

Fig. 18. Comparison of results obtained using different mesh refinement techniques for the stresses at point A: (a) relative error in stresses, (b) related error bounds (GFD are the Green s function decomposition method results in Table 1).

Fig. 12. Results of goal-oriented error estimation for the shell structure shown in Fig. 9: (a) estimated (absolute) relative errors in the quantities of interest and (b) corresponding effectivity indices. The numbers of degrees of freedom refer to the uniformMITC4 element meshes used for the complete shell.

Fig. 13. Predicted absolute pointwise accuracy of the influence function normalized with the quantity of interest defined in (165): body loads in the z-direction in the grey region yield an error in the quantity of interest smaller than a tolerance of (a) tol = 1.0%, (b) tol = 0.8%, (c) tol = 0.6%, (d) tol = 0.4%, (e) tol = 0.2%, (f) tol = 0.1%.

Fig. 2. One-dimensional schematic example for the superconvergence property of the recovered gradient.

Fig. 4. Exact solutions for the bar with varying cross-section A1 = t Æ A2 (E = 1, l = 1, A2 = 1, f = 1).

Fig. 5. Global and local errors for t = 0.01 (E = 1, l = 1, A2 = 1, f = 1). Eight element solution.

Fig. 6. Global error with varying cross-section A1 = t Æ A2 (E = 1, l = 1, A2 = 1, f = 1). Eight element solution.

Frequently asked questions

Q1. What are the contributions in "A posteriori error estimation techniques in practical finite element analysis" ?

In this paper the authors review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. While the authors show how these error estimation techniques are employed for their simple model problem, the emphasis of the paper is on assessing whether these procedures are ready for use in practical linear finite element analysis. The authors conclude that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care. The more accurate estimation procedures also do not provide proven bounds that, in general, can be computed efficiently. 

Q2. What are the future works mentioned in the paper "A posteriori error estimation techniques in practical finite element analysis" ?

Future research work should address the development of actually implementable and practically useful improved error estimators that are applicable for a large class of problems. However, such error bound solution will likely be very difficult to achieve without a significant computational expense. A quite promising approach is to use goal-oriented error measures in order to establish a coarse but still appropriate mesh in computationally intensive finite ele- ment solutions, notably in multi-physics and multi-scale analyses involving optimization. 

Q3. What is the method for estimating error?

Since all the error estimators do not provide in general guaranteed bounds, the estimation technique most useful is probably the method that works efficiently in general analyses (including nonlinear analyses) and for general finite element discretizations, and provides sufficient or reasonable accuracy in the error estimation. 

Q4. What is the key for estimating the error in such quantities?

The key for estimating the error in such quantities is the formulation of an auxiliary problem, which is the dual problem to the primal problem actually considered, and which filters out the necessary information for an accurate estimate for the error in the quantity of interest. 

Q5. What is the error in the shell finite element approximation of the primal problem?

Eh denotes the error in the shell finite element approximation of the primal problem, Zref is the exact influence function corresponding to a 3D reference domain Xref, Zh denotes the finite element solution for Zref obtained on a 3D computational domain Xh, and F are body loads. 

Q6. What is the weak point of the Zienkiewicz–Zhu algorithm?

Another weak point of the Zienkiewicz–Zhu algorithm is the implicit assumption that oscillations indicate errors and that smooth stresses mean accurate stresses. 

Q7. What is the key requirement for an error estimator to be useful in engineering practice?

Of course, a key requirement for this error estimator to be useful in engineering practice is that the computational cost of the error estimate (and ideally the error bounds) must be much smaller than the added computational cost to simply use a very fine mesh.