Abstract
This chapter presents a sufficient condition for convergence in the finite element method for any solution of finite energy. Oliveira carried out a fundamental investigation into the problem of convergence in the finite element method. He also showed that, if a finite element family of fields satisfies a general completeness criterion, then it is complete in energy for a sufficiently smooth subset of the set of functions of finite energy, and that, subject to some further restrictions on the exact solution, the sequence of finite element approximate solutions, obtained by minimizing the functional over the finite element fields, converges in energy to the exact solution. After a function space, formalism on the partitioned domain is developed in a slightly different manner from that of Oliveira, it is shown that a finite element family of fields that satisfies the completeness criterion is, in fact, complete in energy for all functions of finite energy, and that, when in addition the finite element family satisfies a conformity criterion, which is quite weak and usually met in practice, the sequence of approximate solutions always converges in energy to any exact solution of finite energy.
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