Finite Element Analysis - Barna Szabó

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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

      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

      (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

      (1.116)normal upper Phi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis minus normal upper Phi left-parenthesis u Subscript upper F upper E Baseline right-parenthesis equals minus 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 period

      Therefore the error in the extracted data is

      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.

      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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