$P_1$--Nonconforming Quadrilateral Finite Element Space with Periodic Boundary Conditions: Part I. Fundamental results on dimensions, bases, solvers, and error analysis

TL;DR

This paper studies the properties of a specific nonconforming quadrilateral finite element space with periodic boundary conditions, providing fundamental results on its structure, numerical schemes, and error analysis, including extensions to 3D.

Contribution

It characterizes the space's dimension and basis, proposes numerical schemes for elliptic problems, and extends the analysis to three dimensions, addressing challenges with non-invertible matrices.

Findings

Dimension and basis characterized using discrete boundary conditions

Numerical schemes guaranteed to have solutions via Drazin inverse

Extension of results to three-dimensional problems

Abstract

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.