Direct computation of in the point x0 involves the computation of the corresponding standard coordinate using eq. (1.83) and evaluating the following expression:
(1.85)
where . The computation of the higher derivatives is analogous.
Remark 1.8 When plotting quantities of interest such as the functions and , the data for the plotting routine are generated by subdividing the standard element into n intervals of equal length, n being the desired resolution. The QoIs corresponding to the grid‐points are evaluated. This process does not involve inverse mapping. In node points information is provided from the two elements that share that node. If the computed QoI is discontinuous then the discontinuity will be visible at the nodes unless the plotting algorithm automatically averages the QoIs.
Indirect computation of in node points
The first derivative in node points can be determined indirectly from the generalized formulation. For example, to compute the first derivative at node xk from the finite element solution, we select and use
(1.86)
Test functions used in post‐solution operations for the computation of a functional are called extraction functions. Here is an extraction function for the functional . This is because and and hence
(1.87)
where, by definition; .
Example 1.8 Let us find for the problem in Example 1.7 by the direct and indirect methods. In this case the exact solution is known from which we have . By direct computation:
