A nonconforming generalized finite element method for transmission problems

We obtain quasi-optimal rates of convergence for transmission (interface) problems on domains with smooth, curved boundaries using a nonconforming generalized finite element method (GFEM). More precisely, we study the strongly elliptic problem Pu := -∑∂_j (A^ij∂ _iu) = f in a smooth bounded domain Ω with Dirichlet boundary conditions. The coefficients A^ij are piecewise smooth, possibly with jump discontinuities along a smooth, closed surface Γ, called the interface, which does not intersect the boundary of the domain. We consider a sequence of approximation spaces S_μ satisfying two conditions-(1) nearly zero boundary and interface matching and (2) approximability-which are similar to those in Babuška, Nistor, and Tarfulea, [J. Comput. Appl. Math., 218 (2008), pp. 175-183]. Then, if u_μ ∈ S _μ, μ ≥ 1, is a sequence of Galerkin approximations of the solution u to the interface problem, the approximation error ∥u-u _μ∥ _Ĥ1(Ω) is of order O(h _μ^m), where h_μ^m is the typical size of the elements in S^μ and Ĥ¹ is the Sobolev space of functions in H¹ on each side of the interface. We give an explicit construction of GFEM spaces S_μ for which our two assumptions are satisfied, and hence for which the quasi-optimal rates of convergence hold, and present a numerical test.