Eigenvalue Error Expansion And Extrapolation With Extensional Rannacher-Turek Element

This thesis mainly discusses numerical experiment of the error expansion and extrapolation theory of Extensional Rannacher-Turek nonconforming element. Comparing to the other extensively used element, we clearly see which element is more effective. Furthermore, we also devise numerical experiments for Crouziex-Raviart element, which has not theoretical estimation yet. The result shows that Extensional Rannacher-Turek element approximates the true eigenvalue from lower bound as the theory predicts it, which is superior to the linear element. Through extrapolation, the order is poromted from 2 to 4. For Rannacher-Turek element, its single root also gives lower-bound approximation. However, it can reach fourth order and further extrapolation leads to sixth order, while its repeted root has only second order precision and extrapolation leads to fourth order precision. As is guessed, Crouziex-Raviart element gives lower-bound approximation with second order precision and further extrapolation reaches fourth order precision.In the end, this thesis also devises numerical experiment for problems with small perturbation on its boundary strip. Result shows that Rannacher-Turek element becomes exceptionally unstable. On the contary, Extensional Rannacher-Turek element is more reliable. Numerical experiment is not the direct proof to theory, but its approximation to the fact is trustful. Thus, it provides important clues for further theory research.