RECENT DEVELOPMENT OF IMMERSED FEM FOR ELLIPTIC AND ELASTIC INTERFACE PROBLEMS

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We survey a recently developed immersed finite element method (IFEM) for the interface problems. The IFEM uses structured grids such as uniform grids, even if the interface is a smooth curve. Instead of fitting the curved interface, the bases are modified so that they satisfy the jump conditions along the interface. The early versions of IFEM [1, 2] were suboptimal in convergence order [3]. Later, the consistency terms were added to the bilinear forms [4, 5], thus the scheme became optimal and the error estimates were proven. For elasticity problems with interfaces, we modify the Crouzeix-Raviart based element to satisfy the traction conditions along the interface [6], but the consistency terms are not needed. To satisfy the Korn's inequality, we add the stabilizing terms to the bilinear form. The optimal error estimate was shown for a triangular grid. Lastly, we describe the multigrid algorithms for the discretized system arising from IFEM. The prolongation operators are designed so that the prolongated function satisfy the flux continuity condition along the interface. The W-cycle convergence was proved, and the number of V-cycle is independent of the mesh size.
Publisher
KOREAN SOC INDUSTRIAL & APPLIED MATHEMATICS
Issue Date
2019-06
Language
English
Article Type
Article
Citation

JOURNAL OF THE KOREAN SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS, v.23, no.2, pp.65 - 92

ISSN
1226-9433
DOI
10.12941/jksiam.2019.23.065
URI
http://hdl.handle.net/10203/267406
Appears in Collection
MA-Journal Papers(저널논문)
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