Finite Element Analysis. Barna Szabó

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Название Finite Element Analysis
Автор произведения Barna Szabó
Жанр Физика
Серия
Издательство Физика
Год выпуска 0
isbn 9781119426462



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StartAbsoluteValue pi overbar Subscript upper E upper X Superscript left-parenthesis k right-parenthesis Baseline EndAbsoluteValue EndRoot EndFraction dot"/>

      The error of approximation over the entire domain is:

      By Theorem 1.2, the exact value of the potential energy is

      (1.131)pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis equals minus one half integral Subscript 0 Superscript 1 Baseline left-parenthesis u prime Subscript upper E upper X right-parenthesis squared d x equals minus one half left-parenthesis StartFraction alpha squared Over 2 alpha minus 1 EndFraction minus left-parenthesis alpha plus 1 right-parenthesis plus StartFraction left-parenthesis alpha plus 1 right-parenthesis squared Over 2 alpha plus 1 EndFraction right-parenthesis

      and the relative error in energy norm on the entire domain is:

      (1.132)left-parenthesis e Subscript r Baseline right-parenthesis Subscript upper E Baseline equals left-parenthesis StartFraction pi Subscript upper F upper E Baseline minus pi Subscript upper E upper X Baseline Over StartAbsoluteValue pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis EndAbsoluteValue EndFraction right-parenthesis Superscript 1 slash 2 Baseline period

      It is seen that for all values of α the maximum error is associated with the first element.

      The error of approximation for alpha equals 1 is zero. This follows directly from Theorem 1.4: The exact solution is a polynomial of degree 2. Therefore it lies in the finite element space and hence the finite element solution is the same as the exact solution.

Element number
α 1 2 3 4 5