二阶椭圆特征值问题的一种新型混合元格式

A New Mixed Finite Element Scheme for Second Order Elliptic Eigenvalue Problem

为满足实际问题对速度较低的正则性要求,本文建立了二阶椭圆特征值问题的一种新型混合元格式.由于速度空间只需满足平方可积性质,因此混合元配对变得简单易取.本文采用由分片常数速度元和分片线性压力元构成的协调有限元配对,得到椭圆特征值问题的最优误差估计.与传统的混合元配对格式比较,新方法只需较少的自由度便可达到同样的数值精度.最后,数值试验结果与理论分析相吻合,表明新方法的有效性.

In this paper, we propose a new mixed finite element scheme to the second order elliptic eigenvalue problem based on the less regularity of the velocity in practice. Because in the new mixed finite element scheme, the velocity space satisfies square integrable properties, the choices of mixed element pairs become simple and easy. Based on this new scheme, we address corresponding conforming finite element pairs, which consists of piecewise constant element for velocity and piecewise linear element for pressure. Moreover, optimal error estimate of eigenvalues is obtained. Our method needs less degrees of freedom to obtain the same precision compared with classical mixed element schemes. Finally, we give numerical experiments to verify our theoretical results.