Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer

Zhilin Li, Hayk Mikayelyan

Research output: Journal PublicationArticlepeer-review


A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mapping z = √z in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.

Original languageEnglish
Pages (from-to)75-90
Number of pages16
JournalInterfaces and Free Boundaries
Issue number1
Publication statusPublished - 2016


  • Augmented immersed interface method
  • Crack-tip
  • Fast/Poisson solver
  • Free boundary
  • Free discontinuity
  • Irregular domain
  • Mumford-Shah energy

ASJC Scopus subject areas

  • Surfaces and Interfaces


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