【作者】 刘力军；

【导师】 林群；

【作者基本信息】 河北大学 ， 基础数学， 2003， 硕士

【摘要】 本文主要讨论了特征值问题拓广Rannacher-Turek非协调元的误差展开与外推的数值实验结果，与其它几种广泛使用的有限元比较，我们能清楚的看到哪个有限元更有效。本文还对尚未得到理论结果的Crouziex-Raviart有限元进行了数值实验。实验结果表明拓广Rannacher-Turek元确实给出特征值的下逼近，并且要优于线性元，外推可以使2阶精度提高到4阶，而Rannacher-Turek元对单根同样给出下逼近，但近似解本身便可达到4阶高精度，外推可以获得更高阶(6阶)精度，重根却给出上逼近，只具有2阶精度，但可以通过外推达到4阶精度。而Crouziex-Raviart非协调元则如预测，给出下逼近，具有2阶精度，外推可以获得4阶精度。 最后，本文又对所处理区域的边界带有小扰动的问题，进行了数值实验，实验结果表明Rannacher—Turek元变得异常不稳定，而相比之下拓广的Rannacher—Turek元更可靠。数值实验结果虽然不是理论的直接证明，但是其近似真实情况的可靠性是值得信赖的，这无疑对进一步的理论研究提供了重要线索。更多还原

【Abstract】 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.更多还原