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

We consider a mixed-boundary-value/interface problem for the elliptic operator P=-Σ_ij^∂i(aij∂ju)=f on a polygonal domain ΩâŠ^R2 with straight sides. We endowed the boundary of Ω partially with Dirichlet boundary conditions u=0 on ^∂DΩ, and partially with Neumann boundary conditions Σ_ij^νiaij∂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 s_n that yield "^hm-quasi-optimal rates of convergence", m≥1, for the Galerkin approximations u_nε s_n of the solution u. More precisely, we prove that ||-u-u_n||-≤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(s_n)→∞. 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 s_n are also Generalized Finite Element spaces.