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A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces
Author(s) -
Mourad Hashem M.,
Dolbow John,
Harari Isaac
Publication year - 2006
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1788
Subject(s) - lagrange multiplier , finite element method , bubble , constraint algorithm , dirichlet distribution , simple (philosophy) , multiplier (economics) , mathematics , domain (mathematical analysis) , mathematical analysis , computer science , mathematical optimization , boundary value problem , engineering , structural engineering , philosophy , epistemology , parallel computing , economics , macroeconomics
We examine a bubble‐stabilized finite element method for enforcing Dirichlet constraints on embedded interfaces. By ‘embedded’ we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh. As such, the robust imposition of Dirichlet constraints using a Lagrange multiplier field is not trivial. To focus issues, we consider a simple one‐sided problem that is representative of a wide class of evolving‐interface problems. The bulk field is decomposed into coarse and fine scales, giving rise to coarse‐scale and fine‐scale one‐sided sub‐problems. The fine‐scale solution is approximated with bubble functions, permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method. Importantly, the method is simple to implement, readily extends to multiple dimensions, obviates the need to specify any free stabilization parameters, and can lead to a symmetric, positive‐definite system of equations. The performance of the method is then examined through several numerical examples. The accuracy of the Lagrange multiplier is compared to results obtained using a local version of the domain integral method. The variational multiscale approach proposed herein is shown to both stabilize the Lagrange multiplier and improve the accuracy of the post‐processed fluxes. Copyright © 2006 John Wiley & Sons, Ltd.