Finite Element Analysis. Barna Szabó

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      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

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)СКАЧАТЬ