Finite Element Analysis. Barna Szabó

such that

      (1.162)

      We now ask: In what sense will be close to ? The answer is that there is a constant C, independent of the finite element mesh and , such that

(1.163)

      provided, however, that and were properly selected.

For example, let S be the space defined in eq. (1.61) with , . The space has the dimension . For consider three choices:

      1  is the set of functions which are constant on each finite element. has the dimension .

      2  is the space S defined in (3.11) with , (dimension ).

      3  is the set of functions which are linear on every element and discontinuous at the nodes (dimension ).

For these choices of the mixed formulation leads to systems of linear equations with , and unknowns, respectively. In the cases and , a constant C exists such that the inequality (1.163) holds for all , and . In the case of , however, such a constant does not exist. This means that no matter how large C is, there exist some and and mesh so that the inequality (1.163) is not satisfied. On the other hand, there will be and for which the inequality is satisfied and therefore the finite element solutions will converge to the underlying exact solution.

      1.8.2 Nitsche's method

Nitsche's method¹⁶ allows the treatment of essential boundary conditions as natural boundary conditions. This has certain advantages in two and three dimensions. An outline of the algorithmic aspects of the method is presented in the following. For additional details we refer to [51].

      Consider the problem:

(1.164)

      with the boundary conditions and . However, at we substitute the natural boundary condition:

(1.165)

Скачать книгу