Finite Element Analysis. Barna Szabó

      We define a function such that

      (1.112)

      This operation projects onto the space . Letting we get:

      We will write this as

(1.113)

      Next we define such that

      (1.114)

      This operation projects onto the space . By Galerkin's orthogonality condition (see Theorem 1.3) we have

Therefore, letting , we write eq. (1.113) as

      (1.115)

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 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 is of comparable magnitude to 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 , 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