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.
1
North Carolina State University, Raleigh, USA
2
The University of Nottingham Ningbo University, China
Zhilin Li; Hayk Mikayelyan. Fine numerical analysis of the crack-tip position for a Mumford–Shah minimizer. Interfaces and free boundaries, Tome 18 (2016) no. 1, pp. 75-90. doi: 10.4171/ifb/357
@article{10_4171_ifb_357,
author = {Zhilin Li and Hayk Mikayelyan},
title = {Fine numerical analysis of the crack-tip position for a {Mumford{\textendash}Shah} minimizer},
journal = {Interfaces and free boundaries},
pages = {75--90},
year = {2016},
volume = {18},
number = {1},
doi = {10.4171/ifb/357},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/357/}
}
TY - JOUR
AU - Zhilin Li
AU - Hayk Mikayelyan
TI - Fine numerical analysis of the crack-tip position for a Mumford–Shah minimizer
JO - Interfaces and free boundaries
PY - 2016
SP - 75
EP - 90
VL - 18
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/357/
DO - 10.4171/ifb/357
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