Finite Element Analysis. Barna Szabó

      Numerical example

      Letting , , and we construct the numerical problem using one element and the hierarchic shape functions defined in Section 1.3.1. By definition:

      (1.173)

Table 1.7 The computed values of

.

γ	10−3	10−6	10−9	10−12	10−15
u(ℓ)	0.2540348	0.2500004	0.25(0)64	0.25(0)94	0.25(0)124

      where p is the polynomial degree. Therefore and and, using the Legendre shape functions, for the unconstrained coefficient matrix, without the modifications of Nitsche, is

Referring to eq. (1.172), the coefficient matrix is modified by the application of Nitsche's method. Those modifications in the present case are:

      The unconstrained right hand side vector without the modifications of Nitsche is:

      and with the modifications of Nitsche it is:

The numerical results shown in Table 1.7 indicate that the stabilized formulation is remarkably robust. The notation indicates that there are n zeros.

Notes

      1 1 Ludwig Prandtl 1875–1953.

      2 2 The generalized form is also called variational form or weak form.

      3 3 The term “discretization” refers to processes by which approximating functions are defined. The most widely used discretizations will be described and illustrated by examples in this and subsequent chapters.

      4 4 See Definition A.5 in the appendix.

      5 5 Peter Gustav Lejeune Dirichlet 1805–1859.

      6 6 Carl Gottfried Neumann 1832–1925.

      7 7 Victor Gustave Robin (1855–1897).

      8 8 A functional is a real‐valued function defined on a space of functions or vectors.

      9