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Finite element solutions for three‐dimensional elliptic boundary value problems on unbounded domains
Author(s) -
Oh HaeSoo,
Yun JaeHeon,
Jang Bong Soo
Publication year - 2006
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20151
Subject(s) - mathematics , bounded function , finite element method , domain (mathematical analysis) , boundary value problem , boundary (topology) , poincaré–steklov operator , mathematical analysis , complement (music) , partial differential equation , method of fundamental solutions , gravitational singularity , ball (mathematics) , boundary knot method , boundary element method , free boundary problem , robin boundary condition , structural engineering , biochemistry , chemistry , complementation , phenotype , gene , engineering
The finite element (FE) solutions of a general elliptic equation −div([ a ij ] ⋅∇ u ) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R 3 , is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω ∞ , so that the remaining part Ω B = Ω\ Ω ∞ is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γ a = Ω ∞ ∩ Ω B . In this article, instead of discarding an unbounded subdomain Ω ∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babuška and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f . © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006