P1$$ {P}_1 $$-Nonconforming quadrilateral finite element space with periodic boundary conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis

The P1$$ {P}_1 $$-nonconforming quadrilateral finite element space with periodic boundary conditions is investigated. The dimension and basis for the space are characterized by using the concept of minimally essential discrete boundary conditions. We show that the situation is different based on the parity of the number of discretizations on coordinates. Based on the analysis on the space, we propose several numerical schemes for elliptic problems with periodic boundary conditions. Some of these numerical schemes are related to solving linear equations consisting of non-invertible matrices. 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 dimensions is provided.

Automated summary

The P1 nonconforming quadrilateral finite element space with periodic boundary conditions is studied. The dimension and basis of the space are determined using the concept of minimally essential discrete boundary conditions. The situation is found to be different based on the parity of the number of discretizations on coordinates. Several numerical schemes are proposed for solving elliptic problems with periodic boundary conditions, some of which involve solving linear equations with non-invertible matrices. The existence of corresponding numerical solutions is guaranteed using the Drazin inverse. The theoretical relationship between the numerical solutions is derived and confirmed by numerical results. Finally, the extension to three dimensions is provided.