P1$$ {P}_1 $$‐nonconforming quadrilateral finite element space with periodic boundary conditions: Part II. Application to the nonconforming heterogeneous multiscale method

Abstract

A homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such process numerically. In this paper we introduce a FEHMM scheme for multiscale elliptic problems based on nonconforming elements. In particular we use the noconforming element with the periodic boundary condition introduced in the companion paper. Theoretical analysis derives a priori error estimates in the standard Sobolev norms. Several numerical results which confirm our analysis are provided.

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

“…This paper aims to investigate the structure of the $$$ {P}_1 $$$–nonconforming quadrilateral finite element spaces with periodic BC thoroughly and to suggest some iterative methods to solve the resulting linear systems based on the idea of Drazin inverse. An application for nonconforming heterogeneous multiscale methods (NcHMM) of $$$ {P}_1 $$$–nonconforming quadrilateral finite element appears in this journal [29].…”