Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

We consider a mixed boundary‐value problem for the Poisson equation in a thick junction Ωε which is the union of a domain Ω0 and a large number of ε—periodically situated thin cylinders. The non‐uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2011-05, Vol.34 (7), p.758-775
Main Authors: Mel'nyk, T. A., Nakvasiuk, Iu. A., Wendland, W. L.
Format: Article
Language:English
Subjects:
Asymptotic properties
Boundaries
boundary integral equations
Boundary value problems
Calculus of variations and optimal control
Convergence
Cylinders
Exact sciences and technology
homogenization
Homogenizing
Integral equations
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, boundary value problems
Sciences and techniques of general use
Signorini-type conditions
thick junction
variational inequalities
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Summary:We consider a mixed boundary‐value problem for the Poisson equation in a thick junction Ωε which is the union of a domain Ω0 and a large number of ε—periodically situated thin cylinders. The non‐uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as ε→0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non‐uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as ε→0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non‐standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in Ω0 and an appropriate postprocessing. The equations in Ω0 finally are also treated with boundary integral equations. Copyright © 2010 John Wiley & Sons, Ltd.
ISSN:0170-4214
1099-1476
1099-1476
DOI:10.1002/mma.1395