Finite Element Analysis. Barna Szabó

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Figure 1.8 Exercise 1.18. Notation.

independently of the polynomial degree pk. For sign convention refer to Fig. 1.8. Consider both thermal and traction loads. This exercise demonstrates that nodal forces are in equilibrium independently of the finite element solution. Therefore equilibrium of nodal forces is not an indicator of the quality of finite element solutions.

1.5 Estimation of error in energy norm

We have seen that the finite element solution minimizes the error in energy norm in the sense of eq. (1.48). It is natural therefore to use the energy norm as a measure of the error of approximation. There are two types of error estimators: (a) A priori estimators that establish the asymptotic rate of convergence of a discretization scheme, given information about the regularity (smoothness) of the exact solution and (b) a posteriori estimators that provide estimates of the error in energy norm for the finite element solution of a particular problem.

      There is a very substantial body of work in the mathematical literature on the a priori estimation of the rate of convergence, given a quantitative measure of the regularity of the exact solution and a sequence of discretizations. The underlying theory is outside of the scope of this book; however, understanding the main results is important for practitioners of finite element analysis. For details we refer to [28, 45, 70, 84].

1.5.1 Regularity

      Let us consider problems the exact solution of which has the functional form

(1.89)

      where is an analytic or piecewise analytic function, see Definition A.1 in the appendix. Our motivation for considering functions in this form is that this family of functions models the singular behavior of solutions of linear elliptic boundary value problems near vertices in polygonal and polyhedral domains. For to be in the energy space, its first derivative must be square integrable on I. Therefore