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 non-conforming Generalized Finite Element Method (GFEM). More precisely, we study the strongly elliptic problem Pu := − ∑ ∂j(A ∂iu) = f in a smooth bounded domain Ω with Dirichlet boundary conditions. The coefficients Aij 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 Sμ satisfying two conditions: (1) nearly zero boundary and interface matching, (2) approximability, which are similar to those in Babuška, Nistor, and Tarfulea, J. Comput. Appl. Math. (2008). Then, if uμ ∈ Sμ, μ ≥ 1, is a sequence of Galerkin approximations of the solution u to the interface problem, the approximation error ‖u− uμ‖Ĥ1(Ω) is of order O(h m μ ), where h m μ is the typical size of the elements in Sμ and Ĥ1 is the Sobolev space of functions in H1 on each side of the interface. We give an explicit construction of GFEM spaces Sμ for which our two assumptions are satisfied, and hence for which the quasi-optimal rates of convergence hold and present a numerical test.