Runge's phenomenon is the problem of oscillations that occur in polynomial interpolation when the data is fit with a polynomial of high degree. Something similar is also observed in higher order Finite elements, widely known as "Spurious Oscillations". Are the two related? If not, then what causes the "Spurious Oscillations"?
I suggest to consider that the source of any oscillation is stated in the well known Godunov theorem. FEM acts like FD, FV, SM and can not overcome this issue without using special non-linear schemes.
In variational approximations, stability is typically ensured by the (uniform) coercivity of the bilinear form which appears in the weak formulation. If the bilinear form happens to be non-coercive along a subspace, then a "spurious" component in this subspace (e.g., a high-frequency oscillation) may arise.
Roughly speaking, avoiding such situation in the discretized problem requires inf-sup type conditions, like the Ladyzhenskaya-Babuska-Brezzi condition.
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