# A simple finite element method for non-divergence form elliptic equation

## Abstract

Here, we develop a simple finite element method for solving second order elliptic equations in non-divergence form by combining least squares concept with discontinuous approximations. This simple method has a symmetric and positive definite system and can be easily analyzed and implemented. We could have also used general meshes with polytopal element and hanging node in the method. We prove that our finite element solution approaches to the true solution when the mesh size approaches to zero. Numerical examples are tested that demonstrate the robustness and flexibility of the method.

- Authors:

- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
- Univ. of Arkansas, Little Rock, AR (United States). Dept. of Mathematics

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1346680

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- International Journal of Numerical Analysis and Modeling

- Additional Journal Information:
- Journal Volume: 14; Journal Issue: 2; Journal ID: ISSN 1705-5105

- Publisher:
- Institute for Scientific Computing and Information

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; finite element methods; non-divergence form elliptic equations; polyhedral meshes

### Citation Formats

```
Mu, Lin, and Ye, Xiu.
```*A simple finite element method for non-divergence form elliptic equation*. United States: N. p., 2017.
Web.

```
Mu, Lin, & Ye, Xiu.
```*A simple finite element method for non-divergence form elliptic equation*. United States.

```
Mu, Lin, and Ye, Xiu. Wed .
"A simple finite element method for non-divergence form elliptic equation". United States.
doi:. https://www.osti.gov/servlets/purl/1346680.
```

```
@article{osti_1346680,
```

title = {A simple finite element method for non-divergence form elliptic equation},

author = {Mu, Lin and Ye, Xiu},

abstractNote = {Here, we develop a simple finite element method for solving second order elliptic equations in non-divergence form by combining least squares concept with discontinuous approximations. This simple method has a symmetric and positive definite system and can be easily analyzed and implemented. We could have also used general meshes with polytopal element and hanging node in the method. We prove that our finite element solution approaches to the true solution when the mesh size approaches to zero. Numerical examples are tested that demonstrate the robustness and flexibility of the method.},

doi = {},

journal = {International Journal of Numerical Analysis and Modeling},

number = 2,

volume = 14,

place = {United States},

year = {Wed Mar 01 00:00:00 EST 2017},

month = {Wed Mar 01 00:00:00 EST 2017}

}

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