# Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

## Abstract

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulationsmore »

- Authors:

- Department of Mathematics, The University of Tennessee, Knoxville, TN 37996 (United States)
- Department of Mathematics, The University of Massachusetts, North Dartmouth, MA 02747 (United States)

- Publication Date:

- OSTI Identifier:
- 22622267

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 334; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; CONVERGENCE; CRYSTAL MODELS; EFFICIENCY; EPITAXY; EQUATIONS; FOURIER TRANSFORMATION; HILBERT SPACE; LAPLACIAN; LIQUID CRYSTALS; NONLINEAR PROBLEMS; PERIODICITY; PHASE TRANSFORMATIONS; THIN FILMS

### Citation Formats

```
Feng, Wenqiang, E-mail: wfeng1@vols.utk.edu, Salgado, Abner J., E-mail: asalgad1@utk.edu, Wang, Cheng, E-mail: cwang1@umassd.edu, and Wise, Steven M., E-mail: swise1@utk.edu.
```*Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2016.12.046.

```
Feng, Wenqiang, E-mail: wfeng1@vols.utk.edu, Salgado, Abner J., E-mail: asalgad1@utk.edu, Wang, Cheng, E-mail: cwang1@umassd.edu, & Wise, Steven M., E-mail: swise1@utk.edu.
```*Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms*. United States. doi:10.1016/J.JCP.2016.12.046.

```
Feng, Wenqiang, E-mail: wfeng1@vols.utk.edu, Salgado, Abner J., E-mail: asalgad1@utk.edu, Wang, Cheng, E-mail: cwang1@umassd.edu, and Wise, Steven M., E-mail: swise1@utk.edu. Sat .
"Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms". United States. doi:10.1016/J.JCP.2016.12.046.
```

```
@article{osti_22622267,
```

title = {Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms},

author = {Feng, Wenqiang, E-mail: wfeng1@vols.utk.edu and Salgado, Abner J., E-mail: asalgad1@utk.edu and Wang, Cheng, E-mail: cwang1@umassd.edu and Wise, Steven M., E-mail: swise1@utk.edu},

abstractNote = {We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.},

doi = {10.1016/J.JCP.2016.12.046},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 334,

place = {United States},

year = {2017},

month = {4}

}