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 := - Sigma partial derivative(j) (A(ij)partial derivative(i)u) = f in a smooth bounded domain Omega with Dirichlet boundary conditions. The coefficients A(ij) are piecewise smooth, possibly with jump discontinuities along a smooth, closed surface Gamma, called the interface, which does not intersect the boundary of the domain. We consider a sequence of approximation spaces S-mu satisfying two conditions-(1) nearly zero boundary and interface matching and (2) approximability-which are similar to those in Babu. ska, Nistor, and Tarfulea, [J. Comput. Appl. Math., 218 (2008), pp. 175-183]. Then, if u(mu) is an element of S-mu, mu >= 1, is a sequence of Galerkin approximations of the solution u to the interface problem, the approximation error parallel to u-u(mu)parallel to((H) over cap1(Omega)) is of order O(h(mu)(m)), where h(mu)(m) is the typical size of the elements in S-mu and (H) over cap (1) is the Sobolev space of functions in H-1 on each side of the interface. We give an explicit construction of GFEM spaces S-mu for which our two assumptions are satisfied, and hence for which the quasi-optimal rates of convergence hold, and present a numerical test.