$P_1$--Nonconforming Quadrilateral Finite Element Space with Periodic Boundary Conditions: Part II. Application to the Nonconforming Heterogeneous Multiscale Method

TL;DR

This paper develops a finite element heterogeneous multiscale method using nonconforming quadrilateral elements with periodic boundary conditions, providing theoretical error estimates and numerical validation for multiscale elliptic problems.

Contribution

It introduces a FEHMM scheme based on nonconforming elements with periodic boundary conditions and derives a priori error estimates for multiscale elliptic problems.

Findings

Theoretical error estimates in Sobolev norms are established.

Numerical results confirm the accuracy of the proposed FEHMM scheme.

The method effectively handles multiscale elliptic problems with periodic structures.

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 spaces. 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.