# A least squares finite element method using Elsasser variables for magnetohydrodynamic equations

Eunjung Lee, Heonkyu Ha, Sang Dong Kim

Research output: Contribution to journalArticle

### Abstract

There are various different forms of magnetohydrodynamic(MHD) equations and they have been studied for years due to its complicated coupling between variables. This paper proposes to use an equivalently transformed MHD equations with Elsasser variables and the least squares finite element method to find the approximation to them. Introducing new variables by combining fluid velocity and magnetic field yields a Navier–Stokes like system. Then the first-order system least squares method using displacement recasts the transformed MHD equations into a system of first order partial differential equations and the Newton's algorithm linearizes the problem. An L2-residual functional is defined to minimize and the unique existence of corresponding weak solution is shown. Finally, the convergence of proposed approximation is analyzed and several numerical examples are presented to verify the theory.

Original language English 599-608 10 Journal of Computational and Applied Mathematics 346 https://doi.org/10.1016/j.cam.2018.06.038 Published - 2019 Jan 15

### Fingerprint

Least-squares Finite Element Method
Magnetohydrodynamic Equations
Magnetohydrodynamics
Finite element method
First-order System
First order differential equation
Approximation
Navier-Stokes
Least Square Method
Velocity Field
Partial differential equations
Weak Solution
Partial differential equation
Magnetic Field
Magnetic fields
Verify
Minimise
Fluid
Numerical Examples
Fluids

### All Science Journal Classification (ASJC) codes

• Computational Mathematics
• Applied Mathematics

### Cite this

title = "A least squares finite element method using Elsasser variables for magnetohydrodynamic equations",
abstract = "There are various different forms of magnetohydrodynamic(MHD) equations and they have been studied for years due to its complicated coupling between variables. This paper proposes to use an equivalently transformed MHD equations with Elsasser variables and the least squares finite element method to find the approximation to them. Introducing new variables by combining fluid velocity and magnetic field yields a Navier–Stokes like system. Then the first-order system least squares method using displacement recasts the transformed MHD equations into a system of first order partial differential equations and the Newton's algorithm linearizes the problem. An L2-residual functional is defined to minimize and the unique existence of corresponding weak solution is shown. Finally, the convergence of proposed approximation is analyzed and several numerical examples are presented to verify the theory.",
author = "Eunjung Lee and Heonkyu Ha and Kim, {Sang Dong}",
year = "2019",
month = "1",
day = "15",
doi = "10.1016/j.cam.2018.06.038",
language = "English",
volume = "346",
pages = "599--608",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",

}

A least squares finite element method using Elsasser variables for magnetohydrodynamic equations. / Lee, Eunjung; Ha, Heonkyu; Kim, Sang Dong.

In: Journal of Computational and Applied Mathematics, Vol. 346, 15.01.2019, p. 599-608.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A least squares finite element method using Elsasser variables for magnetohydrodynamic equations

AU - Lee, Eunjung

AU - Ha, Heonkyu

AU - Kim, Sang Dong

PY - 2019/1/15

Y1 - 2019/1/15

N2 - There are various different forms of magnetohydrodynamic(MHD) equations and they have been studied for years due to its complicated coupling between variables. This paper proposes to use an equivalently transformed MHD equations with Elsasser variables and the least squares finite element method to find the approximation to them. Introducing new variables by combining fluid velocity and magnetic field yields a Navier–Stokes like system. Then the first-order system least squares method using displacement recasts the transformed MHD equations into a system of first order partial differential equations and the Newton's algorithm linearizes the problem. An L2-residual functional is defined to minimize and the unique existence of corresponding weak solution is shown. Finally, the convergence of proposed approximation is analyzed and several numerical examples are presented to verify the theory.

AB - There are various different forms of magnetohydrodynamic(MHD) equations and they have been studied for years due to its complicated coupling between variables. This paper proposes to use an equivalently transformed MHD equations with Elsasser variables and the least squares finite element method to find the approximation to them. Introducing new variables by combining fluid velocity and magnetic field yields a Navier–Stokes like system. Then the first-order system least squares method using displacement recasts the transformed MHD equations into a system of first order partial differential equations and the Newton's algorithm linearizes the problem. An L2-residual functional is defined to minimize and the unique existence of corresponding weak solution is shown. Finally, the convergence of proposed approximation is analyzed and several numerical examples are presented to verify the theory.

UR - http://www.scopus.com/inward/record.url?scp=85051370302&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051370302&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2018.06.038

DO - 10.1016/j.cam.2018.06.038

M3 - Article

AN - SCOPUS:85051370302

VL - 346

SP - 599

EP - 608

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -