TY - JOUR

T1 - Quasi-optimal rates of convergence for the Generalized Finite Element Method in polygonal domains

AU - Mazzucato, Anna L.

AU - Nistor, Victor

AU - Qu, Qingqin

N1 - Funding Information:
A.L.M. and Q.Q. were partially supported by NSF Grant DMS-0708902 , DMS-1009713 . In addition, A.L.M. was partially supported by NSF Grant DMS-1009714 . V.N. was partially supported by the NSF Grants DMS-0713743 , OCI-0749202 , and DMS-1016556 .

PY - 2014/6

Y1 - 2014/6

N2 - We consider a mixed-boundary-value/interface problem for the elliptic operator P=-Σij∂i(aij∂ju)=f on a polygonal domain ΩâŠR2 with straight sides. We endowed the boundary of Ω partially with Dirichlet boundary conditions u=0 on ∂DΩ, and partially with Neumann boundary conditions Σijνiaij∂ju=0 on ∂NΩ. The coefficients aij are piecewise smooth with jump discontinuities across the interface Γ, 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", m≥1, for the Galerkin approximations unε sn of the solution u. More precisely, we prove that ||-u-un||-≤Cdim(s n)-m/2||-f||Hm-1(Ω), where C depends on the data for the problem, but not on f, u, or n and dim(sn)→∞. Our construction is quite general and depends on a choice of a good sequence of approximation spaces Sn′ on a certain subdomain W that is at some distance to the vertices. In case the spaces Sn′ are Generalized Finite Element spaces, then the resulting spaces sn are also Generalized Finite Element spaces.

AB - We consider a mixed-boundary-value/interface problem for the elliptic operator P=-Σij∂i(aij∂ju)=f on a polygonal domain ΩâŠR2 with straight sides. We endowed the boundary of Ω partially with Dirichlet boundary conditions u=0 on ∂DΩ, and partially with Neumann boundary conditions Σijνiaij∂ju=0 on ∂NΩ. The coefficients aij are piecewise smooth with jump discontinuities across the interface Γ, 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", m≥1, for the Galerkin approximations unε sn of the solution u. More precisely, we prove that ||-u-un||-≤Cdim(s n)-m/2||-f||Hm-1(Ω), where C depends on the data for the problem, but not on f, u, or n and dim(sn)→∞. Our construction is quite general and depends on a choice of a good sequence of approximation spaces Sn′ on a certain subdomain W that is at some distance to the vertices. In case the spaces Sn′ are Generalized Finite Element spaces, then the resulting spaces sn are also Generalized Finite Element spaces.

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U2 - 10.1016/j.cam.2013.12.026

DO - 10.1016/j.cam.2013.12.026

M3 - Article

AN - SCOPUS:84892967394

VL - 263

SP - 466

EP - 477

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

ER -