Finite Element Analysis. Barna Szabó

are finite. Therefore can be approximated by Taylor series about any point of the domain . It is known that the error term of a Taylor series truncated at polynomial degree p is bounded by the th derivative of :

      (1.108)

      In the special case when α is an integer and then .

Exercise 1.19 Show how eq. (1.106) is obtained from eq. (1.105). Provide details.

Example 1.10 Let us consider model problems in the form of eq. (1.103) with the following data: , , and exact solutions in the form of eq. (1.104) corresponding to . We will use a sequence of uniform finite element meshes with and assigned to all elements. We are interested in the relationship between the estimated and true relative errors. The computed values of the potential energy and their estimated limit values computed by means of eq. (1.99) are listed in Table 1.3. These are comparable to the exact values of the potential energy listed in Table 1.2. The estimated limit values of the potential energy are denoted by .

With the information provided in Tables 1.2 and 1.3 it is possible to compare the estimated and exact values of the relative error. For example, using eq. (1.100) and

Table 1.3 Example: Computed and estimated values of the potential energy

. Uniform mesh refinement, for all elements.

	N
10	19	−2.17753673	−1.73038992	−1.41382648	−1.17955239
100	199	−2.25079984	−1.74673700	−1.41675042	−1.17984996
1000	1999	−2.29589857	−1.75303348	−1.41745363	−1.17989453
		−2.37254083	−1.75716094	−1.41768637	−1.17990276

the estimated relative error in energy norm for , is: