On Finite Element Methods for Indefinite Variational problems
In this talk, I will show basic results on indefinite variational problems (for example the weak form of the Helmholtz equation). The existence of a unique solution for such problems requires application of the Fredholm alternative and a separate uniqueness proof.
A sufficiently dense mesh is required for a priori convergence results of indefinite problems. The required mesh density will depend on the frequency, which leads to very rapid growth of number of degrees of freedoms required to model high-frequency problems. This phenomenon is called the High-Frequency problem.