Barna Szabó - Finite Element Analysis

and for we have:

(1.106)

where is the Legendre polynomial of degree and eq. (D.10) was used.

Since the exact solution is known, the exact value of the potential energy can be determined for any set of values of α , κ , c and . When κ and c are both constants then

(1.107)

The exact values of the potential energy for the data , and and various values of α are shown in Table 1.2.

Table 1.2 Exact values of the potential energy for , and .

α		α
0.600	−2.3728354978	1.000	−1.0000000000
0.700	−1.7571858289	1.500	−0.5104166667
0.800	−1.4176885916	2.000	−0.3047619048
0.900	−1.1799028822	3.000	−0.1420634921

When α is a fractional number then derivatives higher than α will not be finite in . In the range the first derivative in the point is infinity. This range of α has considerable practical importance because the exact solutions of two‐ and three‐dimensional problems often have analogous terms.

When α is an integer then all derivatives of 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 1.19Show how eq. (1.106)is obtained from eq. (1.105). Provide details.

Example 1.10Let 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 1.2and 1.3it 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.