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      Exercise 1.20 Compare the estimated and exact values of the relative error in energy norm for the problem in Example 1.10 for , .

Example 1.11 Let us consider once again model problems in the form of eq. (1.103) with the data: , , and exact solutions corresponding to , see eq. (1.104). Using a sequence of uniform finite element meshes with and assigned to each element, the results shown in Fig. 1.9 are obtained. The values of β were computed by linear regression using eq. (1.101). We observe that . This is consistent with the asymptotic estimate given by eq. (1.91).

Example 1.12 Let us consider model problems in the form of eq. (1.103) with the data: , , and exact solutions corresponding to , see eq. (1.104). Using a uniform finite element mesh with and assigned to each element, the results shown in Fig. 1.10 are obtained. The values of β were computed by linear regression using eq. (1.101). We observe that , that is, the rate of convergence is twice that in Example 1.11. This is consistent with the theoretical results in [22, 84]: The rate of p‐convergence is at least twice the rate of h‐convergence when the singular point is a nodal point.

1.5.4 Error in the extracted QoI

      In Example 1.9 it was demonstrated that the QoI can be extracted from the finite element solution efficiently and accurately even when the discretization was very poorly chosen. Let us consider a quantity of interest and the corresponding extraction function . The extracted value of the QoI is

(1.109)

Figure 1.9 Relative error in energy norm.

,

.

Figure 1.10 Relative error in energy norm.

,

.

      and the exact value of the QoI is

(1.110)

Subtracting eq. (1.109) from eq. (1.110) we get

(1.111) Скачать книгу