# Higher Order Lagrange Finite Elements In M3D

## Abstract

The M3D code has been using linear finite elements to represent multilevel MHD on 2-D poloidal planes. Triangular higher order elements, up to third order, are constructed here in order to provide M3D the capability to solve highly anisotropic transport problems. It is found that higher order elements are essential to resolve the thin transition layer characteristic of the anisotropic transport equation, particularly when the strong anisotropic direction is not aligned with one of the Cartesian coordinates. The transition layer is measured by the profile width, which is zero for infinite anisotropy. It is shown that only higher order schemes have the ability to make this layer converge towards zero when the anisotropy gets stronger and stronger. Two cases are considered. One has the strong transport direction partially aligned with one of the element edges, the other doesn't have any alignment. Both cases have the strong transport direction misaligned with the grid line by some angles.

- Authors:

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab., Princeton, NJ (US)

- Sponsoring Org.:
- USDOE Office of Science (SC) (US)

- OSTI Identifier:
- 836490

- Report Number(s):
- PPPL-4032

TRN: US0500614

- DOE Contract Number:
- AC02-76CH03073

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: 17 Dec 2004

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; ALIGNMENT; ANISOTROPY; CARTESIAN COORDINATES; TRANSPORT; COMPUTATIONAL PHYSICS; COMPUTER SIMULATION; FINITE ELEMENT METHOD

### Citation Formats

```
J. Chen, H.R. Strauss, S.C. Jardin, W. Park, L.E. Sugiyama, G. Fu, and J. Breslau.
```*Higher Order Lagrange Finite Elements In M3D*. United States: N. p., 2004.
Web. doi:10.2172/836490.

```
J. Chen, H.R. Strauss, S.C. Jardin, W. Park, L.E. Sugiyama, G. Fu, & J. Breslau.
```*Higher Order Lagrange Finite Elements In M3D*. United States. doi:10.2172/836490.

```
J. Chen, H.R. Strauss, S.C. Jardin, W. Park, L.E. Sugiyama, G. Fu, and J. Breslau. Fri .
"Higher Order Lagrange Finite Elements In M3D". United States. doi:10.2172/836490. https://www.osti.gov/servlets/purl/836490.
```

```
@article{osti_836490,
```

title = {Higher Order Lagrange Finite Elements In M3D},

author = {J. Chen and H.R. Strauss and S.C. Jardin and W. Park and L.E. Sugiyama and G. Fu and J. Breslau},

abstractNote = {The M3D code has been using linear finite elements to represent multilevel MHD on 2-D poloidal planes. Triangular higher order elements, up to third order, are constructed here in order to provide M3D the capability to solve highly anisotropic transport problems. It is found that higher order elements are essential to resolve the thin transition layer characteristic of the anisotropic transport equation, particularly when the strong anisotropic direction is not aligned with one of the Cartesian coordinates. The transition layer is measured by the profile width, which is zero for infinite anisotropy. It is shown that only higher order schemes have the ability to make this layer converge towards zero when the anisotropy gets stronger and stronger. Two cases are considered. One has the strong transport direction partially aligned with one of the element edges, the other doesn't have any alignment. Both cases have the strong transport direction misaligned with the grid line by some angles.},

doi = {10.2172/836490},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Fri Dec 17 00:00:00 EST 2004},

month = {Fri Dec 17 00:00:00 EST 2004}

}