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=−∑ij∂i(aij∂ju)=f on a polygonal domain Ω⊂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 ∑ijνiaij∂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 un∈Sn of the solution u. More precisely, we prove that ‖u−un‖≤Cdim(Sn)−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.