کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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6422632 | 1632028 | 2014 | 12 صفحه PDF | دانلود رایگان |
We consider a mixed-boundary-value/interface problem for the elliptic operator P=ââijâi(aijâju)=f on a polygonal domain ΩâR2 with straight sides. We endowed the boundary of Ω partially with Dirichlet boundary conditions u=0 on âDΩ, and partially with Neumann boundary conditions âijνiaijâju=0 on âNΩ. The coefficients aij are piecewise smooth with jump discontinuities across the interface Î, which is allowed to have singularities and cross the boundary of Ω. In particular, we consider “triple-junctions” and even “multiple junctions”. Our main result is to construct a sequence of Generalized Finite Element spaces Sn that yield “hm-quasi-optimal rates of convergence”, mâ¥1, for the Galerkin approximations unâSn of the solution u. More precisely, we prove that âuâunââ¤Cdim(Sn)âm/2âfâHmâ1(Ω), where C depends on the data for the problem, but not on f, u, or n and dim(Sn)ââ. Our construction is quite general and depends on a choice of a good sequence of approximation spaces Snâ² on a certain subdomain W that is at some distance to the vertices. In case the spaces Snâ² are Generalized Finite Element spaces, then the resulting spaces Sn are also Generalized Finite Element spaces.
Journal: Journal of Computational and Applied Mathematics - Volume 263, June 2014, Pages 466-477