BIT Numerical Mathematics

, Volume 53, Issue 3, pp 791–820

# A uniformly well-conditioned, unfitted Nitsche method for interface problems

• Sara Zahedi
• Gunilla Kreiss
• Martin Berggren
Article

## Abstract

A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented. The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and jump conditions at the interface are weakly enforced using a variant of Nitsche’s method. In our method, the solutions on each side of the interface are extended to the entire domain which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. We prove that the method provides optimal convergence order in the energy and the L 2 norms and a condition number of the system matrix that is independent of the position of the interface relative to the mesh. Numerical experiments confirm the theoretical results and demonstrate optimal convergence order also for the pointwise errors.

## Keywords

Interface problem Nitsche’s method Interior penalties Finite element methods

## Mathematics Subject Classification (2000)

35J05 35J20 65N15 65N30

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## Authors and Affiliations

• 1
Email author
• Sara Zahedi
• 2
• Gunilla Kreiss
• 2
• Martin Berggren
• 1
1. 1.Department of Computing ScienceUmeå UniversityUmeåSweden
2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden