An L¹ Penalty Method for General Obstacle Problems
Abstract
We construct an efficient numerical scheme for solving obstacle problems in divergence form. The numerical method is based on a reformulation of the obstacle in terms of an L¹-like penalty on the variational problem. The reformulation is an exact regularizer in the sense that for a large (but finite) penalty parameter, we recover the exact solution. Our formulation is applied to classical elliptic obstacle problems as well as some related free boundary problems, for example, the two-phase membrane problem and the Hele--Shaw model. One advantage of the proposed method is that the free boundary inherent in the obstacle problem arises naturally in our energy minimization without any need for problem specific or complicated discretization. In addition, our scheme also works for nonlinear variational inequalities arising from convex minimization problems.
Additional Information
© 2015 Society for Industrial and Applied Mathematics. Received by the editors April 3, 2014; accepted for publication (in revised form) March 3, 2015; published electronically July 7, 2015. The authors would like to thank Inwon Kim for her helpful discussions. The authors would also like to thank Qiang Du and the anonymous referees for their helpful comments on our first draft. The first author was supported by UC Lab 443948-B1-69763 and Keck Funds 449041-PW-58414. The second author's work was supported by NSF DMS 1300445. The third author's work was supported by ONR Grant N00014-11-1-719. This author's work was supported by NSF 1303892 and University of California Presidents Postdoctoral Fellowship Program.Attached Files
Published - 140963303.pdf
Submitted - 1404.1370v1.pdf
Files
Name | Size | Download all |
---|---|---|
md5:f204753bc897d965895733ed8028c970
|
786.9 kB | Preview Download |
md5:9fa9f5e8b5151e31795ec04bfd2fc63d
|
360.3 kB | Preview Download |
Additional details
- Eprint ID
- 60484
- Resolver ID
- CaltechAUTHORS:20150924-122200084
- 443948-B1-69763
- University of California
- 449041-PW-58414
- W. M. Keck Foundation
- DMS 1300445
- NSF
- N00014-11-1-719
- Office of Naval Research (ONR)
- 1303892
- NSF
- Created
-
2015-09-24Created from EPrint's datestamp field
- Updated
-
2023-03-02Created from EPrint's last_modified field