Finite Element Analysis. Barna Szabó

Finite Element Analysis

B left-parenthesis u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline comma w right-parenthesis period"/>

We define a function z Subscript upper E upper X Baseline element-of upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis such that

(1.112) upper B left-parenthesis z Subscript upper E upper X Baseline comma v right-parenthesis equals upper B left-parenthesis w comma v right-parenthesis for all v element-of upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis period

This operation projects w element-of upper E left-parenthesis upper I right-parenthesis onto the space upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis . Letting v equals u Subscript upper E upper X Baseline minus u Subscript upper F upper E we get:

upper B left-parenthesis z Subscript upper E upper X Baseline comma u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline right-parenthesis equals upper B left-parenthesis w comma u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline right-parenthesis for all v element-of upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis period

We will write this as

(1.113) upper B left-parenthesis u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline comma w right-parenthesis equals upper B left-parenthesis u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline comma z Subscript upper E upper X Baseline right-parenthesis period

Next we define z Subscript upper F upper E Baseline element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis such that

(1.114) upper B left-parenthesis z Subscript upper E upper X Baseline comma v right-parenthesis equals upper B left-parenthesis z Subscript upper F upper E Baseline comma v right-parenthesis for all v element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis period

This operation projects z Subscript upper E upper X Baseline element-of upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis onto the space upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis . By Galerkin's orthogonality condition (see Theorem 1.3) we have

upper B left-parenthesis u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline comma v right-parenthesis equals 0 for all v element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis period

Therefore, letting v equals z Subscript upper F upper E , we write eq. (1.113) as

(1.115) upper B left-parenthesis u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline comma w right-parenthesis equals upper B left-parenthesis u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline comma z Subscript upper E upper X Baseline minus z Subscript upper F upper E Baseline right-parenthesis

and we can write eq. (1.111) as

(1.116)

Therefore the error in the extracted data is

(1.117)

where we used the Schwarz inequality, see Section A.3 in the appendix.

The function z Subscript upper F upper E made it possible to write the error in the QoI in this form. It does not have to be computed.

Inequality (1.117) serves to explain why the error in the extracted data can converge to zero faster than the error in energy norm: If double-vertical-bar z Subscript upper E upper X Baseline minus z Subscript upper F upper E Baseline double-vertical-bar Subscript upper E left-parenthesis upper I right-parenthesis is of comparable magnitude to double-vertical-bar u Subscript upper E upper X Baseline minus u Subscript upper F upper E Baseline double-vertical-bar Subscript upper E left-parenthesis upper I right-parenthesis then the error in the extracted data is of comparable magnitude to the error in the strain energy, that is, the error in energy norm squared. But, as seen in Example 1.9, where w was much smoother than u Subscript upper E upper X , it can be much smaller. In the exceptional case when the extraction function is Green's function, the error is zero.

1.6 The choice of discretization in 1D

In an ideal discretization the error (in energy norm) associated with each element would be the same. This ideal discretization can be approximated by automated adaptive methods in which the discretization is modified based on feedback information from previously obtained finite element solutions. Alternatively, based on a general understanding of the relationship between regularity and discretization, and understanding the strengths and limitations

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