The $P_1$--nonconforming quadrilateral finite element space with periodic boundary condition is investigated. The dimension and basis for the space are characterized with the concept of minimally essential discrete boundary conditions. We show that the situation is totally different based on the parity of the number of discretization on coordinates. Based on the analysis on the space, we propose several numerical schemes for elliptic problems with periodic boundary condition. Some of these numerical schemes are related with solving a linear equation consisting of a non-invertible matrix. By courtesy of the Drazin inverse, the existence of corresponding numerical solutions is guaranteed. The theoretical relation between the numerical solutions is derived, and it is confirmed by numerical results. Finally, the extension to the three dimensional is provided.
翻译:根据对空间的分析,我们针对定期边界条件的离散问题提出了若干数字计划。其中一些数字计划与解决由非垂直矩阵组成的线性方程有关。根据Drazin的推论,可以保证存在相应的数字解决方案。数字解决方案之间的理论关系得到推导,并得到数字结果的确认。最后,提供了三维的扩展。