Finite Element Analysis. Barna Szabó

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Finite Element Analysis - Barna Szabó

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v right-parenthesis f o r a l l v element-of upper H 0 Superscript 1 Baseline left-parenthesis upper I right-parenthesis comma upper G element-of upper L squared left-parenthesis upper I right-parenthesis period"/>

      The finite element problem is formulated as follows: Find u Subscript upper F upper E Baseline element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis where upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis is a subspace of upper H 0 Superscript 1 Baseline left-parenthesis upper I right-parenthesis and upper F Subscript upper F upper E Baseline element-of upper V left-parenthesis upper I right-parenthesis where upper V left-parenthesis upper I right-parenthesis is a subspace of upper L squared left-parenthesis upper I right-parenthesis such that

      (1.162)upper B left-parenthesis u Subscript upper F upper E Baseline comma upper F Subscript upper F upper E Baseline semicolon v comma upper G right-parenthesis equals upper F left-parenthesis v right-parenthesis f o r a l l v element-of upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis comma upper G element-of upper V left-parenthesis upper I right-parenthesis period

      We now ask: In what sense will left-parenthesis u Subscript upper F upper E Baseline comma upper F Subscript upper F upper E Baseline right-parenthesis be close to left-parenthesis u Subscript upper E upper X Baseline comma upper F Subscript upper E upper X Baseline right-parenthesis? The answer is that there is a constant C, independent of the finite element mesh and left-parenthesis u Subscript upper E upper X Baseline comma upper F Subscript upper E upper X Baseline right-parenthesis, such that

      provided, however, that upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis and upper V left-parenthesis upper I right-parenthesis were properly selected.

      1  is the set of functions which are constant on each finite element. has the dimension .

      2  is the space S defined in (3.11) with , (dimension ).

      3  is the set of functions which are linear on every element and discontinuous at the nodes (dimension ).

      1.8.2 Nitsche's method

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