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 ∑ 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 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‖ ≤ C dim(Sn)‖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 S′ n on a certain subdomain W that is at some distance to the vertices. In case the spaces S′ n are Generalized Finite Element spaces, then the resulting spaces Sn are also Generalized Finite Element spaces.