Mixed Generalized Multiscale Finite Element Method for a Simplified Magnetohydrodynamics Problem in Perforated Domains

Valentin Alekseev,Qili Tang,Yalchin Efendiev,Maria Vasilyeva,Eric T Chung

doi:10.3390/computation8020058

Abstract

In this paper, we consider a coupled system of equations that describes simplified magnetohydrodynamics (MHD) problem in perforated domains. We construct a fine grid that resolves the perforations on the grid level in order to use a traditional approximation. For the solution on the fine grid, we construct approximation using the mixed finite element method. To reduce the size of the fine grid system, we will develop a Mixed Generalized Multiscale Finite Element Method (Mixed GMsFEM). The method differs from existing approaches and requires some modifications to represent the flow and magnetic fields. Numerical results are presented for a two-dimensional model problem in perforated domains. This model problem is a special case for the general 3D problem. We study the influence of the number of multiscale basis functions on the accuracy of the method and show that the proposed method provides a good accuracy with few basis functions.

Highlights

The magnetohydrodynamic (MHD) system describes the interactions of electrically conducting incompressible flows in the presence of a magnetic field
The presented work is based on our previous papers [30,31], where we presented a Mixed Generalized Multiscale Finite Element Methods (GMsFEM) for solutions of the problems in perforated domains
We present numerical results for the model problem in the perforated domain

Summary

Introduction

The magnetohydrodynamic (MHD) system describes the interactions of electrically conducting incompressible flows in the presence of a magnetic field. There are some research works devoted to the numerical approximation of MHD equations. For treating the nonlinear terms effectively, three classical iterative methods (Stokes-type, Newtown and Oseen-type ones) within finite element approximation for the steady MHD equations have been developed in [4,5]. The theoretical analysis and numerical examples in [4,6] show that the Picard iteration (called Oseen-type iteration by the authors in the reference) is suitable for high Reynolds number problems. The mixed finite element methods with divergence-free velocities and magnetics fields are developed [11,12,13]. Analysis of the above literature mainly focuses on Computation 2020, 8, 58; doi:10.3390/computation8020058 www.mdpi.com/journal/computation

Objectives

Results

Conclusion