Augmented immersed finite element methods for some elliptic partial differential equations

Abstract

Augmented immersed finite element methods are proposed to solve elliptic interface problems with non-homogeneous jump conditions. The non-homogeneous jump conditions are treated as source terms using the singularity removal technique. For the piecewise constant coefficient case, we transform the original interface problem to a Poisson equation with the same jump in the solution, but an unknown flux jump (augmented variable) which is chosen such that the original flux jump condition is satisfied. The GMRES iterative method is used to solve the augmented variable. The core of each iteration involves solving a Poisson equation using a fast Poisson solver and an interpolation scheme to interpolate the flux jump condition. With a little modification, the method can be applied to solve Poisson equations on irregular domains. Numerical experiments show that not only the computed solution but also the normal derivative are second-order accurate in the $L^{\mathrm{\infty }}$ norm.

Keywords:

• interface problem
• non-homogeneous jump condition
• irregular domain
• level set function
• singularity removal technique

2000 AMS Subject Classifications::

• 65N15
• 65N30
• 35J60

Acknowledgments

The authors would like to thank the anonymous reviewers for constructive comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

The first and second authors were supported by the National Science Foundation (NSF) of China [Grants No. 11371199 and 11301275], the Program of Natural Science Research of Jiangsu Higher Education Institutions of China [Grant No. 12KJB110013], and the Doctoral fund of Ministry of Education of China [Grant No. 20123207120001], and the Innovation Project for Graduate Education of Jiangsu Province (CXLX13_365). The third author is partially supported by the US AFSOR grant FA9550-09-1-0520, the NSF grant DMS-0911434, and the NIH grant 5R01GM96195-2, and CNSF grants 10971102, 11471166, and BK20141443.