Finite Element Analysis. Barna Szabó

      On integrating by parts, we get the following expression which is better suited for numerical evaluation:

      (1.123)

      We address the following questions: (a) How does the error in energy norm depend on the parameter α, the mesh Δ and the p‐distribution p? and (b) How is this error distributed among the elements? Understanding these relationships is necessary for making sound choices of discretization based on a priori information concerning the regularity of the exact solution.

      We compute the potential energy of the difference between the exact solution and its linear interpolant for the kth element:

The exact solution for and its linear interpolant for , uniform mesh, are shown in Fig. 1.12.

      To obtain the potential energy of the difference between the exact solution and its linear interpolant for the kth element, denoted by , we need to solve:

      (1.124)

Figure 1.12 The exact solution for

and its linear interpolant for

, uniform mesh.

      The solution is

      (1.125)

      Using eq. (1.106) we get

      (1.126)

      where is the difference between and its linear interpolant:

      (1.127)

      and compute

      Referring to Theorem 1.5, the error in energy norm associated with the kth element is

(1.128)

      and the relative error in energy norm associated with the kth element is:

(1.129)

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