### Table 1 A posteriori error estimates and bounds for the asymptotic behaviour of the addi- tional terms. Algorithm jjj(u ? uh; p ? ph)jjj

"... In PAGE 20: ... Thus, the extra terms have to be computed in practice to guarantee an asymptotically correct a posteriori error estimate. Table1 shows that the asymptotic behaviour of the additional terms in all a posteriori error estimates decreases by k powers of H if the Newton step is replaced by the Oseen step. The additional correction step improves the asymptotic order of convergence of the extra terms in the a posteriori error estimates for the L2{norm of the velocity by the factor H1? quot; and it does not in uence the asymptotic behaviour of the extra terms in the a posteriori error estimates in the jjj jjj{norm.... ..."

### Table 1: The a posteriori error estimates Acknowledgement: The author wishes to acknowledge Prof. W. Layton for fruitful discussions on the subject of this paper and for the critical reading of the manuscript. References

"... In PAGE 20: ...solving the Navier{Stokes equations on a mesh TH with H gt; h, performing one Newton Step on the mesh Th and solving a defect correction equation on TH . The behaviour of the a posteriori error estimators in comparison to the estimators for Algo- rithm 1 is presented in Table1 where 1 = 1((uh; ph)) and 2 = 2((uh; ph)). Here is quot; gt; 0 for d = 2, quot; = 1=2 for d = 3, k = minfpolynomial degree of the velocity nite elements, polynomial degree of the pressure nite elements + 1, smoothness of the solution (1:6)g and the scaling between the grids as assumed in the previous sections.... ..."

### Table 5: Example 8.2, exact and estimated global L2{error using ^ . In Table 5 we present results using the local a posteriori error estimator ^ from (6.5). The jumps of the nonconforming function are weighted by the factor 1. This term dominates in ^ . Obviously, the order of convergence of this term is similar to that of ku ? uhk0. This leads to the fact that ^ is always an upper bound of the global error. In contrast, the quantities Cs^ completely overestimate the error. Following Remark 6.5 we have tested a modi ed a posteriori error estimator whose local terms have the form

1998

"... In PAGE 29: ... Following Remark 6.5 we have tested a modi ed a posteriori error estimator whose local terms have the form 2 T =h4 T kf+ uh?b ruh?cuhk2 0;T+X E @Th3 Ek [j ruh nEj]E k2 0;E+hEk( +hEkbk1;E) [juhj]E k2 0;E: The behaviour of ^ and di ers considerably, compare Table5 and 6. The jump terms of the nonconforming function are not longer dominant.... ..."

Cited by 5

### Table 5: Example 8.2, exact and estimated global L2{error using ^ . In Table 5 we present results using the local a posteriori error estimator ^ from (6.5). The jumps of the nonconforming function are weighted by the factor 1. This term dominates in ^ . Obviously, the order of convergence of this term is similar to that of ku ? uhk0. This leads to the fact that ^ is always an upper bound of the global error. In contrast, the quantities Cs^ completely overestimate the error. Following Remark 6.5 we have tested a modi ed a posteriori error estimator whose local terms have the form

1998

"... In PAGE 29: ... Following Remark 6.5 we have tested a modi ed a posteriori error estimator whose local terms have the form 2 T =h4 T kf+ uh?b ruh?cuhk2 0;T+X E @Th3 Ek [j ruh nEj]E k2 0;E+hEk( +hEkbk1;E) [juhj]E k2 0;E: The behaviour of ^ and di ers considerably, compare Table5 and 6. The jump terms of the nonconforming function are not longer dominant.... ..."

Cited by 5

### Tables 16 and 17 show that the a posteriori error estimate works well in the nonsmooth case. Note, in particular, in Table 17 how the percentage of the grid which fails the paraboloid test is approximately halved with the grid spacing. In Figure 4, we have plotted the viscosity solution to the nonsmooth test problem on an 80 80 grid, the absolute value of the error and the region which failed the paraboloid test. Note the rapid variation in the error in the region of the kinks, corresponding to the singularity of the exact solution. Note also how it corresponds well to the singularities in the true solution. These preliminary results indicate that it is reasonable to expect that in several dimensions the a posteriori error estimate will behave in a manner similar to that observed in one dimension.

1995

Cited by 12

### Table 1: Results obtained for computing u(0) for u = sin( (2x1 + x2 + 2)), via the energy and L2 error estimators H1 and L2 compared to the point error estimator loc mesh obtained for our sample solution we see that one would hardly have come up with such a mesh by mere heuristic reasoning. The above concept for deriving a posteriori error estimates L # n e(0)

1995

Cited by 18

### Table 1: Example 7.1, error and convergence rate in k k0 for uniform re nement. An improvement of these results is only possible with a better resolution of the layers. That apos;s why a local a posteriori error estimator should generate grids where above all the region of the 21

1998

"... In PAGE 21: ... The solution and the coarse grid (level 0) are presented in Figure 5. The results using uniform re nement are given in Table1 where the convergence rate is computed using the results of the level 6 and 7.... ..."

Cited by 5

### Table 1: Example 7.1, error and convergence rate in k k0 for uniform re nement. An improvement of these results is only possible with a better resolution of the layers. That apos;s why a local a posteriori error estimator should generate grids where above all the region of the 21

1998

"... In PAGE 21: ... The solution and the coarse grid (level 0) are presented in Figure 5. The results using uniform re nement are given in Table1 where the convergence rate is computed using the results of the level 6 and 7.... ..."

Cited by 5

### Table 1: Convergence of the e ectivity indices for the three adaptive analyses. 4. REFERENCES [1] J. Barlow. Optimal Stress Locations in Finite Element Models. Int. J. Numer. Meths. Engrg., 10, 243{251 (1976). [2] T. Blacker and T. Belytschko. Superconvergent Patch Recovery with Equilibrium and Conjoint Inter- polant Enhancements. Int. J. Numer. Meths. Engrg., 37, 517{536 (1994). [3] N.-E. Wiberg, F. Abdulwahab, and S. Ziukas. Enhanced Superconvergent Patch Recovery Incorporat- ing Equilibrium and Boundary Conditions. Int. J. Numer. Meths. Engrg., 37, 3417{3440 (1994). [4] O. C. Zienkiewicz and J. Z. Zhu. The Superconvergent Patch Recovery and a posteriori Error Estimates. Part 1: The Recovery Technique. Int. J. Numer. Meths. Engrg., 33, 1331{1364 (1992). 4

"... In PAGE 4: ... In both of the latter analyses a bi-linear polynomial is assumed for the recovered stress eld over the patch of elements. In Table1 the convergence of the e ectivity indices, 1 and 2, are reported for the three analyses. These indices are de ned through 1 = v u u t 1 nel nel X e=1 ke ek keek ? 1 2 ; 2 = ke k kek where kek = v u u t nel X e=1 keek2 (14) ke ek and keek denote the energy norms of the estimated and exact errors, respectively, for element number e.... ..."