## Abstract

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}∂ _{i}u) = 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 Ĥ^{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_{μ} for which our two assumptions are satisfied, and hence for which the quasi-optimal rates of convergence hold, and present a numerical test.

Original language | English (US) |
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Pages (from-to) | 555-576 |

Number of pages | 22 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

## All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics